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Found 102 declarations mentioning CategoryTheory.Functor.uncurry.

CategoryTheory.Functor.uncurry 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : CategoryTheory.Functor (CategoryTheory.Functor C (CategoryTheory.Functor D E)) (CategoryTheory.Functor (C × D) E)
CategoryTheory.Functor.fullyFaithfulUncurry 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : CategoryTheory.Functor.uncurry.FullyFaithful
CategoryTheory.Functor.instFaithfulProdUncurry 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : CategoryTheory.Functor.uncurry.Faithful
CategoryTheory.Functor.instFullProdUncurry 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] : CategoryTheory.Functor.uncurry.Full
CategoryTheory.Functor.uncurry_obj_obj 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : C × D) : (CategoryTheory.Functor.uncurry.obj F).obj X = (F.obj X.1).obj X.2
CategoryTheory.Functor.uncurry_obj_curry_obj 📋 Mathlib.CategoryTheory.Functor.Currying
{B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] (F : CategoryTheory.Functor (B × C) D) : CategoryTheory.Functor.uncurry.obj (CategoryTheory.Functor.curry.obj F) = F
CategoryTheory.Functor.curry_obj_uncurry_obj 📋 Mathlib.CategoryTheory.Functor.Currying
{B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] (F : CategoryTheory.Functor B (CategoryTheory.Functor C D)) : CategoryTheory.Functor.curry.obj (CategoryTheory.Functor.uncurry.obj F) = F
CategoryTheory.Functor.uncurry_obj_injective 📋 Mathlib.CategoryTheory.Functor.Currying
{B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {F₁ F₂ : CategoryTheory.Functor B (CategoryTheory.Functor C D)} (h : CategoryTheory.Functor.uncurry.obj F₁ = CategoryTheory.Functor.uncurry.obj F₂) : F₁ = F₂
CategoryTheory.Functor.uncurryObjFlip 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) : CategoryTheory.Functor.uncurry.obj F.flip ≅ (CategoryTheory.Prod.swap D C).comp (CategoryTheory.Functor.uncurry.obj F)
CategoryTheory.Functor.flipIsoCurrySwapUncurry 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) : F.flip ≅ CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap D C).comp (CategoryTheory.Functor.uncurry.obj F))
CategoryTheory.Functor.comp_flip_uncurry_eq 📋 Mathlib.CategoryTheory.Functor.Currying
{B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor B D) (G : CategoryTheory.Functor D (CategoryTheory.Functor C E)) : CategoryTheory.Functor.uncurry.obj (F.comp G).flip = ((CategoryTheory.Functor.id C).prod F).comp (CategoryTheory.Functor.uncurry.obj G.flip)
CategoryTheory.Functor.compFlipUncurryIso 📋 Mathlib.CategoryTheory.Functor.Currying
{B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor B D) (G : CategoryTheory.Functor D (CategoryTheory.Functor C E)) : CategoryTheory.Functor.uncurry.obj (F.comp G).flip ≅ ((CategoryTheory.Functor.id C).prod F).comp (CategoryTheory.Functor.uncurry.obj G.flip)
CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip 📋 Mathlib.CategoryTheory.Functor.Currying
{B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {H : Type u₅} [CategoryTheory.Category.{v₅, u₅} H] (F₁ : CategoryTheory.Functor B C) (F₂ : CategoryTheory.Functor D E) (G : CategoryTheory.Functor (C × E) H) : CategoryTheory.Functor.uncurry.obj (F₂.comp (F₁.comp (CategoryTheory.Functor.curry.obj G)).flip).flip = (F₁.prod F₂).comp G
CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip' 📋 Mathlib.CategoryTheory.Functor.Currying
{B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {H : Type u₅} [CategoryTheory.Category.{v₅, u₅} H] (F₁ : CategoryTheory.Functor B C) (F₂ : CategoryTheory.Functor D E) (G : CategoryTheory.Functor (C × E) H) : CategoryTheory.Functor.uncurry.obj (F₁.comp (F₂.comp (CategoryTheory.Functor.curry.obj G).flip).flip) = (F₁.prod F₂).comp G
CategoryTheory.Functor.uncurry_obj_map 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) {X Y : C × D} (f : X ⟶ Y) : (CategoryTheory.Functor.uncurry.obj F).map f = CategoryTheory.CategoryStruct.comp ((F.map f.1).app X.2) ((F.obj Y.1).map f.2)
CategoryTheory.Functor.uncurryObjFlip_hom_app 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D × C) : F.uncurryObjFlip.hom.app X = CategoryTheory.CategoryStruct.id ((F.obj X.2).obj X.1)
CategoryTheory.Functor.uncurryObjFlip_inv_app 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D × C) : F.uncurryObjFlip.inv.app X = CategoryTheory.CategoryStruct.id ((F.obj X.2).obj X.1)
CategoryTheory.Functor.compFlipUncurryIso_hom_app 📋 Mathlib.CategoryTheory.Functor.Currying
{B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor B D) (G : CategoryTheory.Functor D (CategoryTheory.Functor C E)) (X : C × B) : (F.compFlipUncurryIso G).hom.app X = CategoryTheory.CategoryStruct.id ((G.obj (F.obj X.2)).obj X.1)
CategoryTheory.Functor.compFlipUncurryIso_inv_app 📋 Mathlib.CategoryTheory.Functor.Currying
{B : Type u₁} [CategoryTheory.Category.{v₁, u₁} B] {C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor B D) (G : CategoryTheory.Functor D (CategoryTheory.Functor C E)) (X : C × B) : (F.compFlipUncurryIso G).inv.app X = CategoryTheory.CategoryStruct.id ((G.obj (F.obj X.2)).obj X.1)
CategoryTheory.Functor.flipIsoCurrySwapUncurry_hom_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D) (X✝ : C) : (F.flipIsoCurrySwapUncurry.hom.app X).app X✝ = CategoryTheory.CategoryStruct.id ((F.obj X✝).obj X)
CategoryTheory.Functor.flipIsoCurrySwapUncurry_inv_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (F : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X : D) (X✝ : C) : (F.flipIsoCurrySwapUncurry.inv.app X).app X✝ = CategoryTheory.CategoryStruct.id ((F.obj X✝).obj X)
CategoryTheory.Functor.whiskeringRight₂_obj_obj_obj_map 📋 Mathlib.CategoryTheory.Functor.Currying
(B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X✝ : CategoryTheory.Functor B C) (Y : CategoryTheory.Functor B D) {X✝¹ Y✝ : B} (f : X✝¹ ⟶ Y✝) : ((((CategoryTheory.Functor.whiskeringRight₂ B C D E).obj X).obj X✝).obj Y).map f = CategoryTheory.CategoryStruct.comp ((X.map (X✝.map f)).app (Y.obj X✝¹)) ((X.obj (X✝.obj Y✝)).map (Y.map f))
CategoryTheory.Functor.currying_counitIso_hom_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor (C × D) E) (X✝ : C × D) : (CategoryTheory.Functor.currying.counitIso.hom.app X).app X✝ = CategoryTheory.CategoryStruct.id (X.obj (X✝.1, X✝.2))
CategoryTheory.Functor.currying_counitIso_inv_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor (C × D) E) (X✝ : C × D) : (CategoryTheory.Functor.currying.counitIso.inv.app X).app X✝ = CategoryTheory.CategoryStruct.id (X.obj (X✝.1, X✝.2))
CategoryTheory.Functor.uncurry_map_app 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] {X✝ Y✝ : CategoryTheory.Functor C (CategoryTheory.Functor D E)} (T : X✝ ⟶ Y✝) (X : C × D) : (CategoryTheory.Functor.uncurry.map T).app X = (T.app X.1).app X.2
CategoryTheory.Functor.currying_unitIso_hom_app_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X✝ : C) (X✝¹ : D) : ((CategoryTheory.Functor.currying.unitIso.hom.app X).app X✝).app X✝¹ = CategoryTheory.CategoryStruct.id ((X.obj X✝).obj X✝¹)
CategoryTheory.Functor.currying_unitIso_inv_app_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
{C : Type u₂} [CategoryTheory.Category.{v₂, u₂} C] {D : Type u₃} [CategoryTheory.Category.{v₃, u₃} D] {E : Type u₄} [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X✝ : C) (X✝¹ : D) : ((CategoryTheory.Functor.currying.unitIso.inv.app X).app X✝).app X✝¹ = CategoryTheory.CategoryStruct.id ((X.obj X✝).obj X✝¹)
CategoryTheory.Functor.whiskeringRight₂_obj_obj_map_app 📋 Mathlib.CategoryTheory.Functor.Currying
(B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) (X✝ : CategoryTheory.Functor B C) {X✝¹ Y✝ : CategoryTheory.Functor B D} (g : X✝¹ ⟶ Y✝) (X✝² : B) : ((((CategoryTheory.Functor.whiskeringRight₂ B C D E).obj X).obj X✝).map g).app X✝² = (X.obj (X✝.obj X✝²)).map (g.app X✝²)
CategoryTheory.Functor.whiskeringRight₂_map_app_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
(B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] {X✝ Y✝ : CategoryTheory.Functor C (CategoryTheory.Functor D E)} (f : X✝ ⟶ Y✝) (X : CategoryTheory.Functor B C) (Y : CategoryTheory.Functor B D) (c : B) : ((((CategoryTheory.Functor.whiskeringRight₂ B C D E).map f).app X).app Y).app c = (f.app (X.obj c)).app (Y.obj c)
CategoryTheory.Functor.whiskeringRight₂_obj_map_app_app 📋 Mathlib.CategoryTheory.Functor.Currying
(B : Type u₁) [CategoryTheory.Category.{v₁, u₁} B] (C : Type u₂) [CategoryTheory.Category.{v₂, u₂} C] (D : Type u₃) [CategoryTheory.Category.{v₃, u₃} D] (E : Type u₄) [CategoryTheory.Category.{v₄, u₄} E] (X : CategoryTheory.Functor C (CategoryTheory.Functor D E)) {X✝ Y✝ : CategoryTheory.Functor B C} (f : X✝ ⟶ Y✝) (Y : CategoryTheory.Functor B D) (X✝¹ : B) : ((((CategoryTheory.Functor.whiskeringRight₂ B C D E).obj X).map f).app Y).app X✝¹ = (X.map (f.app X✝¹)).app (Y.obj X✝¹)
CategoryTheory.Limits.colimitIsoSwapCompColim 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
{C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasColimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) : CategoryTheory.Limits.colimit G ≅ (CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp (CategoryTheory.Functor.uncurry.obj G))).comp CategoryTheory.Limits.colim
CategoryTheory.Limits.limitIsoSwapCompLim 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
{C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasLimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) : CategoryTheory.Limits.limit G ≅ (CategoryTheory.Functor.curry.obj ((CategoryTheory.Prod.swap K J).comp (CategoryTheory.Functor.uncurry.obj G))).comp CategoryTheory.Limits.lim
CategoryTheory.Limits.colimitIsoSwapCompColim_hom_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
{C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasColimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) (X : K) : (CategoryTheory.Limits.colimitIsoSwapCompColim G).hom.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation G X).hom (CategoryTheory.Limits.colimMap (G.flipIsoCurrySwapUncurry.hom.app X))
CategoryTheory.Limits.colimitIsoSwapCompColim_inv_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
{C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasColimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) (X : K) : (CategoryTheory.Limits.colimitIsoSwapCompColim G).inv.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimMap (G.flipIsoCurrySwapUncurry.inv.app X)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation G X).inv
CategoryTheory.Limits.limitIsoSwapCompLim_hom_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
{C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasLimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) (X : K) : (CategoryTheory.Limits.limitIsoSwapCompLim G).hom.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation G X).hom (CategoryTheory.Limits.limMap (G.flipIsoCurrySwapUncurry.hom.app X))
CategoryTheory.Limits.limitIsoSwapCompLim_inv_app 📋 Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
{C : Type u} [CategoryTheory.Category.{v, u} C] {J : Type u₁} [CategoryTheory.Category.{v₁, u₁} J] {K : Type u₂} [CategoryTheory.Category.{v₂, u₂} K] [CategoryTheory.Limits.HasLimitsOfShape J C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) (X : K) : (CategoryTheory.Limits.limitIsoSwapCompLim G).inv.app X = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limMap (G.flipIsoCurrySwapUncurry.inv.app X)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation G X).inv
CategoryTheory.MonoidalCategory.externalProductBifunctor_map_app 📋 Mathlib.CategoryTheory.Monoidal.ExternalProduct.Basic
(J₁ : Type u₁) (J₂ : Type u₂) (C : Type u₃) [CategoryTheory.Category.{v₁, u₁} J₁] [CategoryTheory.Category.{v₂, u₂} J₂] [CategoryTheory.Category.{v₃, u₃} C] [CategoryTheory.MonoidalCategory C] {X Y : CategoryTheory.Functor J₁ C × CategoryTheory.Functor J₂ C} (f : X ⟶ Y) (X✝ : J₁ × J₂) : ((CategoryTheory.MonoidalCategory.externalProductBifunctor J₁ J₂ C).map f).app X✝ = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight (f.1.app X✝.1) (X.2.obj X✝.2)) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft (Y.1.obj X✝.1) (f.2.app X✝.2))
CategoryTheory.Limits.coconeOfCoconeUncurry 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCocones F} (Q : (j : J) → CategoryTheory.Limits.IsColimit (D.obj j)) (c : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj F)) : CategoryTheory.Limits.Cocone D.coconePoints
CategoryTheory.Limits.coneOfConeUncurry 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCones F} (Q : (j : J) → CategoryTheory.Limits.IsLimit (D.obj j)) (c : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj F)) : CategoryTheory.Limits.Cone D.conePoints
CategoryTheory.Limits.colimitUncurryIsoColimitCompColim 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasColimitsOfShape K C] [CategoryTheory.Limits.HasColimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasColimit (F.comp CategoryTheory.Limits.colim)] : CategoryTheory.Limits.colimit (CategoryTheory.Functor.uncurry.obj F) ≅ CategoryTheory.Limits.colimit (F.comp CategoryTheory.Limits.colim)
CategoryTheory.Limits.limitUncurryIsoLimitCompLim 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasLimitsOfShape K C] [CategoryTheory.Limits.HasLimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasLimit (F.comp CategoryTheory.Limits.lim)] : CategoryTheory.Limits.limit (CategoryTheory.Functor.uncurry.obj F) ≅ CategoryTheory.Limits.limit (F.comp CategoryTheory.Limits.lim)
CategoryTheory.Limits.coconeOfCoconeUncurryIsColimit 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCocones F} (Q : (j : J) → CategoryTheory.Limits.IsColimit (D.obj j)) {c : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj F)} (P : CategoryTheory.Limits.IsColimit c) : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.coconeOfCoconeUncurry Q c)
CategoryTheory.Limits.coneOfConeUncurryIsLimit 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCones F} (Q : (j : J) → CategoryTheory.Limits.IsLimit (D.obj j)) {c : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj F)} (P : CategoryTheory.Limits.IsLimit c) : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.coneOfConeUncurry Q c)
CategoryTheory.Limits.IsColimit.ofCoconeUncurry 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCocones F} (Q : (j : J) → CategoryTheory.Limits.IsColimit (D.obj j)) {c : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj F)} (P : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.coconeOfCoconeUncurry Q c)) : CategoryTheory.Limits.IsColimit c
CategoryTheory.Limits.IsLimit.ofConeOfConeUncurry 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCones F} (Q : (j : J) → CategoryTheory.Limits.IsLimit (D.obj j)) {c : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj F)} (P : CategoryTheory.Limits.IsLimit (CategoryTheory.Limits.coneOfConeUncurry Q c)) : CategoryTheory.Limits.IsLimit c
CategoryTheory.Limits.coconeOfCoconeUncurry_pt 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCocones F} (Q : (j : J) → CategoryTheory.Limits.IsColimit (D.obj j)) (c : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj F)) : (CategoryTheory.Limits.coconeOfCoconeUncurry Q c).pt = c.pt
CategoryTheory.Limits.coneOfConeUncurry_pt 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCones F} (Q : (j : J) → CategoryTheory.Limits.IsLimit (D.obj j)) (c : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj F)) : (CategoryTheory.Limits.coneOfConeUncurry Q c).pt = c.pt
CategoryTheory.Limits.limitUncurryIsoLimitCompLim_hom_π_π 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasLimitsOfShape K C] [CategoryTheory.Limits.HasLimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasLimit (F.comp CategoryTheory.Limits.lim)] {j : J} {k : K} : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limitUncurryIsoLimitCompLim F).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Limits.lim) j) (CategoryTheory.Limits.limit.π (F.obj j) k)) = CategoryTheory.Limits.limit.π (CategoryTheory.Functor.uncurry.obj F) (j, k)
CategoryTheory.Limits.colimitUncurryIsoColimitCompColim_ι_ι_inv 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasColimitsOfShape K C] [CategoryTheory.Limits.HasColimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasColimit (F.comp CategoryTheory.Limits.colim)] {j : J} {k : K} : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.obj j) k) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.comp CategoryTheory.Limits.colim) j) (CategoryTheory.Limits.colimitUncurryIsoColimitCompColim F).inv) = CategoryTheory.Limits.colimit.ι (CategoryTheory.Functor.uncurry.obj F) (j, k)
CategoryTheory.Limits.colimitUncurryIsoColimitCompColim_ι_hom 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasColimitsOfShape K C] [CategoryTheory.Limits.HasColimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasColimit (F.comp CategoryTheory.Limits.colim)] {j : J} {k : K} : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (CategoryTheory.Functor.uncurry.obj F) (j, k)) (CategoryTheory.Limits.colimitUncurryIsoColimitCompColim F).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.obj j) k) (CategoryTheory.Limits.colimit.ι (F.comp CategoryTheory.Limits.colim) j)
CategoryTheory.Limits.limitUncurryIsoLimitCompLim_inv_π 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasLimitsOfShape K C] [CategoryTheory.Limits.HasLimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasLimit (F.comp CategoryTheory.Limits.lim)] {j : J} {k : K} : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limitUncurryIsoLimitCompLim F).inv (CategoryTheory.Limits.limit.π (CategoryTheory.Functor.uncurry.obj F) (j, k)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Limits.lim) j) (CategoryTheory.Limits.limit.π (F.obj j) k)
CategoryTheory.Limits.limitUncurryIsoLimitCompLim_hom_π_π_assoc 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasLimitsOfShape K C] [CategoryTheory.Limits.HasLimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasLimit (F.comp CategoryTheory.Limits.lim)] {j : J} {k : K} {Z : C} (h : (F.obj j).obj k ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limitUncurryIsoLimitCompLim F).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Limits.lim) j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (F.obj j) k) h)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (CategoryTheory.Functor.uncurry.obj F) (j, k)) h
CategoryTheory.Limits.colimitUncurryIsoColimitCompColim_ι_ι_inv_assoc 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasColimitsOfShape K C] [CategoryTheory.Limits.HasColimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasColimit (F.comp CategoryTheory.Limits.colim)] {j : J} {k : K} {Z : C} (h : CategoryTheory.Limits.colimit (CategoryTheory.Functor.uncurry.obj F) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.obj j) k) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.comp CategoryTheory.Limits.colim) j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimitUncurryIsoColimitCompColim F).inv h)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (CategoryTheory.Functor.uncurry.obj F) (j, k)) h
CategoryTheory.Limits.limitUncurryIsoLimitCompLim_inv_π_assoc 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasLimitsOfShape K C] [CategoryTheory.Limits.HasLimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasLimit (F.comp CategoryTheory.Limits.lim)] {j : J} {k : K} {Z : C} (h : (CategoryTheory.Functor.uncurry.obj F).obj (j, k) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limitUncurryIsoLimitCompLim F).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (CategoryTheory.Functor.uncurry.obj F) (j, k)) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Limits.lim) j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (F.obj j) k) h)
CategoryTheory.Limits.colimitUncurryIsoColimitCompColim_ι_hom_assoc 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] (F : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasColimitsOfShape K C] [CategoryTheory.Limits.HasColimit (CategoryTheory.Functor.uncurry.obj F)] [CategoryTheory.Limits.HasColimit (F.comp CategoryTheory.Limits.colim)] {j : J} {k : K} {Z : C} (h : CategoryTheory.Limits.colimit (F.comp CategoryTheory.Limits.colim) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (CategoryTheory.Functor.uncurry.obj F) (j, k)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimitUncurryIsoColimitCompColim F).hom h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.obj j) k) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (F.comp CategoryTheory.Limits.colim) j) h)
CategoryTheory.Limits.coconeOfCoconeUncurry_ι_app 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCocones F} (Q : (j : J) → CategoryTheory.Limits.IsColimit (D.obj j)) (c : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj F)) (j : J) : (CategoryTheory.Limits.coconeOfCoconeUncurry Q c).ι.app j = (Q j).desc { pt := c.pt, ι := { app := fun k => c.ι.app (j, k), naturality := ⋯ } }
CategoryTheory.Limits.coneOfConeUncurry_π_app 📋 Mathlib.CategoryTheory.Limits.Fubini
{J : Type u_1} {K : Type u_2} [CategoryTheory.Category.{v_1, u_1} J] [CategoryTheory.Category.{v_2, u_2} K] {C : Type u_3} [CategoryTheory.Category.{v_3, u_3} C] {F : CategoryTheory.Functor J (CategoryTheory.Functor K C)} {D : CategoryTheory.Limits.DiagramOfCones F} (Q : (j : J) → CategoryTheory.Limits.IsLimit (D.obj j)) (c : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj F)) (j : J) : (CategoryTheory.Limits.coneOfConeUncurry Q c).π.app j = (Q j).lift { pt := c.pt, π := { app := fun k => c.π.app (j, k), naturality := ⋯ } }
CategoryTheory.Functor.mapCocone₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (c₁ : CategoryTheory.Limits.Cocone K₁) (c₂ : CategoryTheory.Limits.Cocone K₂) : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))
CategoryTheory.Functor.mapCone₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (c₁ : CategoryTheory.Limits.Cone K₁) (c₂ : CategoryTheory.Limits.Cone K₂) : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))
CategoryTheory.Limits.instHasColimitProdObjFunctorUncurryWhiskeringLeft₂OfPreservesColimit₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.HasColimit K₁] [CategoryTheory.Limits.HasColimit K₂] [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] : CategoryTheory.Limits.HasColimit (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))
CategoryTheory.Limits.instHasLimitProdObjFunctorUncurryWhiskeringLeft₂OfPreservesLimit₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.HasLimit K₁] [CategoryTheory.Limits.HasLimit K₂] [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] : CategoryTheory.Limits.HasLimit (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))
CategoryTheory.Limits.isColimitOfPreserves₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cocone K₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {c₂ : CategoryTheory.Limits.Cocone K₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) : CategoryTheory.Limits.IsColimit (G.mapCocone₂ c₁ c₂)
CategoryTheory.Limits.isLimitOfPreserves₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cone K₁} (hc₁ : CategoryTheory.Limits.IsLimit c₁) {c₂ : CategoryTheory.Limits.Cone K₂} (hc₂ : CategoryTheory.Limits.IsLimit c₂) : CategoryTheory.Limits.IsLimit (G.mapCone₂ c₁ c₂)
CategoryTheory.Limits.PreservesColimit₂.mk 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} {G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)} (nonempty_isColimit_mapCocone₂ : ∀ {c₁ : CategoryTheory.Limits.Cocone K₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {c₂ : CategoryTheory.Limits.Cocone K₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂), Nonempty (CategoryTheory.Limits.IsColimit (G.mapCocone₂ c₁ c₂))) : CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G
CategoryTheory.Limits.PreservesColimit₂.nonempty_isColimit_mapCocone₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} {inst✝ : CategoryTheory.Category.{v_1, u_1} J₁} {inst✝¹ : CategoryTheory.Category.{v_2, u_2} J₂} {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} {inst✝² : CategoryTheory.Category.{v_3, u_3} C₁} {inst✝³ : CategoryTheory.Category.{v_4, u_4} C₂} {inst✝⁴ : CategoryTheory.Category.{v_5, u_5} C} {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} {G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)} [self : CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cocone K₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {c₂ : CategoryTheory.Limits.Cocone K₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) : Nonempty (CategoryTheory.Limits.IsColimit (G.mapCocone₂ c₁ c₂))
CategoryTheory.Limits.PreservesLimit₂.mk 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} {G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)} (nonempty_isLimit_mapCone₂ : ∀ {c₁ : CategoryTheory.Limits.Cone K₁} (hc₁ : CategoryTheory.Limits.IsLimit c₁) {c₂ : CategoryTheory.Limits.Cone K₂} (hc₂ : CategoryTheory.Limits.IsLimit c₂), Nonempty (CategoryTheory.Limits.IsLimit (G.mapCone₂ c₁ c₂))) : CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G
CategoryTheory.Limits.PreservesLimit₂.nonempty_isLimit_mapCone₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} {inst✝ : CategoryTheory.Category.{v_1, u_1} J₁} {inst✝¹ : CategoryTheory.Category.{v_2, u_2} J₂} {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} {inst✝² : CategoryTheory.Category.{v_3, u_3} C₁} {inst✝³ : CategoryTheory.Category.{v_4, u_4} C₂} {inst✝⁴ : CategoryTheory.Category.{v_5, u_5} C} {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} {G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)} [self : CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cone K₁} (hc₁ : CategoryTheory.Limits.IsLimit c₁) {c₂ : CategoryTheory.Limits.Cone K₂} (hc₂ : CategoryTheory.Limits.IsLimit c₂) : Nonempty (CategoryTheory.Limits.IsLimit (G.mapCone₂ c₁ c₂))
CategoryTheory.Functor.mapCocone₂_pt 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (c₁ : CategoryTheory.Limits.Cocone K₁) (c₂ : CategoryTheory.Limits.Cocone K₂) : (G.mapCocone₂ c₁ c₂).pt = (G.obj c₁.pt).obj c₂.pt
CategoryTheory.Functor.mapCone₂_pt 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (c₁ : CategoryTheory.Limits.Cone K₁) (c₂ : CategoryTheory.Limits.Cone K₂) : (G.mapCone₂ c₁ c₂).pt = (G.obj c₁.pt).obj c₂.pt
CategoryTheory.Limits.PreservesColimit₂.isoColimitUncurryWhiskeringLeft₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasColimit K₁] [CategoryTheory.Limits.HasColimit K₂] : CategoryTheory.Limits.colimit (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) ≅ (G.obj (CategoryTheory.Limits.colimit K₁)).obj (CategoryTheory.Limits.colimit K₂)
CategoryTheory.Limits.PreservesLimit₂.isoLimitUncurryWhiskeringLeft₂ 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasLimit K₁] [CategoryTheory.Limits.HasLimit K₂] : CategoryTheory.Limits.limit (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) ≅ (G.obj (CategoryTheory.Limits.limit K₁)).obj (CategoryTheory.Limits.limit K₂)
CategoryTheory.Limits.PreservesColimit₂.isoObjCoconePointsOfIsColimit 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cocone K₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {c₂ : CategoryTheory.Limits.Cocone K₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) {c₃ : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsColimit c₃) : (G.obj c₁.pt).obj c₂.pt ≅ c₃.pt
CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsLimit 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cone K₁} (hc₁ : CategoryTheory.Limits.IsLimit c₁) {c₂ : CategoryTheory.Limits.Cone K₂} (hc₂ : CategoryTheory.Limits.IsLimit c₂) {c₃ : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsLimit c₃) : (G.obj c₁.pt).obj c₂.pt ≅ c₃.pt
CategoryTheory.Functor.mapCocone₂_ι_app 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (c₁ : CategoryTheory.Limits.Cocone K₁) (c₂ : CategoryTheory.Limits.Cocone K₂) (x✝ : J₁ × J₂) : (G.mapCocone₂ c₁ c₂).ι.app x✝ = CategoryTheory.CategoryStruct.comp ((G.map (c₁.ι.app x✝.1)).app (K₂.obj x✝.2)) ((G.obj c₁.pt).map (c₂.ι.app x✝.2))
CategoryTheory.Functor.mapCone₂_π_app 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (c₁ : CategoryTheory.Limits.Cone K₁) (c₂ : CategoryTheory.Limits.Cone K₂) (x✝ : J₁ × J₂) : (G.mapCone₂ c₁ c₂).π.app x✝ = CategoryTheory.CategoryStruct.comp ((G.map (c₁.π.app x✝.1)).app c₂.pt) ((G.obj (K₁.obj x✝.1)).map (c₂.π.app x✝.2))
CategoryTheory.Limits.PreservesLimit₂.isoLimitUncurryWhiskeringLeft₂_hom_comp_map_π 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasLimit K₁] [CategoryTheory.Limits.HasLimit K₂] (j : J₁ × J₂) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesLimit₂.isoLimitUncurryWhiskeringLeft₂ K₁ K₂ G).hom (CategoryTheory.CategoryStruct.comp ((G.map (CategoryTheory.Limits.limit.π K₁ j.1)).app (CategoryTheory.Limits.limit K₂)) ((G.obj (K₁.obj j.1)).map (CategoryTheory.Limits.limit.π K₂ j.2))) = CategoryTheory.Limits.limit.π (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) j
CategoryTheory.Limits.PreservesColimit₂.ι_comp_isoColimitUncurryWhiskeringLeft₂_hom 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasColimit K₁] [CategoryTheory.Limits.HasColimit K₂] (j : J₁ × J₂) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) j) (CategoryTheory.Limits.PreservesColimit₂.isoColimitUncurryWhiskeringLeft₂ K₁ K₂ G).hom = CategoryTheory.CategoryStruct.comp ((G.map (CategoryTheory.Limits.colimit.ι K₁ j.1)).app (K₂.obj j.2)) ((G.obj (CategoryTheory.Limits.colimit K₁)).map (CategoryTheory.Limits.colimit.ι K₂ j.2))
CategoryTheory.Limits.PreservesLimit₂.isoLimitUncurryWhiskeringLeft₂_inv_comp_π 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasLimit K₁] [CategoryTheory.Limits.HasLimit K₂] (j : J₁ × J₂) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesLimit₂.isoLimitUncurryWhiskeringLeft₂ K₁ K₂ G).inv (CategoryTheory.Limits.limit.π (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) j) = CategoryTheory.CategoryStruct.comp ((G.map (CategoryTheory.Limits.limit.π K₁ j.1)).app (CategoryTheory.Limits.limit K₂)) ((G.obj (K₁.obj j.1)).map (CategoryTheory.Limits.limit.π K₂ j.2))
CategoryTheory.Limits.PreservesColimit₂.map_ι_comp_isoColimitUncurryWhiskeringLeft₂_inv 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasColimit K₁] [CategoryTheory.Limits.HasColimit K₂] (j : J₁ × J₂) : CategoryTheory.CategoryStruct.comp ((G.map (CategoryTheory.Limits.colimit.ι K₁ j.1)).app (K₂.obj j.2)) (CategoryTheory.CategoryStruct.comp ((G.obj (CategoryTheory.Limits.colimit K₁)).map (CategoryTheory.Limits.colimit.ι K₂ j.2)) (CategoryTheory.Limits.PreservesColimit₂.isoColimitUncurryWhiskeringLeft₂ K₁ K₂ G).inv) = CategoryTheory.Limits.colimit.ι (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) j
CategoryTheory.Limits.PreservesColimit₂.map_ι_comp_isoColimitUncurryWhiskeringLeft₂_inv_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasColimit K₁] [CategoryTheory.Limits.HasColimit K₂] (j : J₁ × J₂) {Z : C} (h : CategoryTheory.Limits.colimit (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) ⟶ Z) : CategoryTheory.CategoryStruct.comp ((G.map (CategoryTheory.Limits.colimit.ι K₁ j.1)).app (K₂.obj j.2)) (CategoryTheory.CategoryStruct.comp ((G.obj (CategoryTheory.Limits.colimit K₁)).map (CategoryTheory.Limits.colimit.ι K₂ j.2)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesColimit₂.isoColimitUncurryWhiskeringLeft₂ K₁ K₂ G).inv h)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) j) h
CategoryTheory.Limits.PreservesLimit₂.isoLimitUncurryWhiskeringLeft₂_hom_comp_map_π_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasLimit K₁] [CategoryTheory.Limits.HasLimit K₂] (j : J₁ × J₂) {Z : C} (h : (G.obj (K₁.obj j.1)).obj (K₂.obj j.2) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesLimit₂.isoLimitUncurryWhiskeringLeft₂ K₁ K₂ G).hom (CategoryTheory.CategoryStruct.comp ((G.map (CategoryTheory.Limits.limit.π K₁ j.1)).app (CategoryTheory.Limits.limit K₂)) (CategoryTheory.CategoryStruct.comp ((G.obj (K₁.obj j.1)).map (CategoryTheory.Limits.limit.π K₂ j.2)) h)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) j) h
CategoryTheory.Limits.PreservesColimit₂.ι_comp_isoColimitUncurryWhiskeringLeft₂_hom_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasColimit K₁] [CategoryTheory.Limits.HasColimit K₂] (j : J₁ × J₂) {Z : C} (h : (G.obj (CategoryTheory.Limits.colimit K₁)).obj (CategoryTheory.Limits.colimit K₂) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesColimit₂.isoColimitUncurryWhiskeringLeft₂ K₁ K₂ G).hom h) = CategoryTheory.CategoryStruct.comp ((G.map (CategoryTheory.Limits.colimit.ι K₁ j.1)).app (K₂.obj j.2)) (CategoryTheory.CategoryStruct.comp ((G.obj (CategoryTheory.Limits.colimit K₁)).map (CategoryTheory.Limits.colimit.ι K₂ j.2)) h)
CategoryTheory.Limits.PreservesLimit₂.isoLimitUncurryWhiskeringLeft₂_inv_comp_π_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] (K₁ : CategoryTheory.Functor J₁ C₁) (K₂ : CategoryTheory.Functor J₂ C₂) (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] [CategoryTheory.Limits.HasLimit K₁] [CategoryTheory.Limits.HasLimit K₂] (j : J₁ × J₂) {Z : C} (h : (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)).obj j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesLimit₂.isoLimitUncurryWhiskeringLeft₂ K₁ K₂ G).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)) j) h) = CategoryTheory.CategoryStruct.comp ((G.map (CategoryTheory.Limits.limit.π K₁ j.1)).app (CategoryTheory.Limits.limit K₂)) (CategoryTheory.CategoryStruct.comp ((G.obj (K₁.obj j.1)).map (CategoryTheory.Limits.limit.π K₂ j.2)) h)
CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsColimit_inv_comp_map_π 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cone K₁} (hc₁ : CategoryTheory.Limits.IsLimit c₁) {c₂ : CategoryTheory.Limits.Cone K₂} (hc₂ : CategoryTheory.Limits.IsLimit c₂) {c₃ : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsLimit c₃) (j : J₁ × J₂) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).inv (CategoryTheory.CategoryStruct.comp ((G.map (c₁.π.app j.1)).app c₂.pt) ((G.obj (K₁.obj j.1)).map (c₂.π.app j.2))) = c₃.π.app j
CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsLimit_hom_comp_π 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cone K₁} (hc₁ : CategoryTheory.Limits.IsLimit c₁) {c₂ : CategoryTheory.Limits.Cone K₂} (hc₂ : CategoryTheory.Limits.IsLimit c₂) {c₃ : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsLimit c₃) (j : J₁ × J₂) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).hom (c₃.π.app j) = CategoryTheory.CategoryStruct.comp ((G.map (c₁.π.app j.1)).app c₂.pt) ((G.obj (K₁.obj j.1)).map (c₂.π.app j.2))
CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsColimit_inv_comp_map_π_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cone K₁} (hc₁ : CategoryTheory.Limits.IsLimit c₁) {c₂ : CategoryTheory.Limits.Cone K₂} (hc₂ : CategoryTheory.Limits.IsLimit c₂) {c₃ : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsLimit c₃) (j : J₁ × J₂) {Z : C} (h : (G.obj (K₁.obj j.1)).obj (K₂.obj j.2) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).inv (CategoryTheory.CategoryStruct.comp ((G.map (c₁.π.app j.1)).app c₂.pt) (CategoryTheory.CategoryStruct.comp ((G.obj (K₁.obj j.1)).map (c₂.π.app j.2)) h)) = CategoryTheory.CategoryStruct.comp (c₃.π.app j) h
CategoryTheory.Limits.PreservesColimit₂.map_ι_comp_isoObjConePointsOfIsColimit_hom 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cocone K₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {c₂ : CategoryTheory.Limits.Cocone K₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) {c₃ : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsColimit c₃) (j : J₁ × J₂) : CategoryTheory.CategoryStruct.comp ((G.map (c₁.ι.app j.1)).app (K₂.obj j.2)) (CategoryTheory.CategoryStruct.comp ((G.obj c₁.pt).map (c₂.ι.app j.2)) (CategoryTheory.Limits.PreservesColimit₂.isoObjCoconePointsOfIsColimit G hc₁ hc₂ hc₃).hom) = c₃.ι.app j
CategoryTheory.Limits.PreservesColimit₂.ι_comp_isoObjConePointsOfIsColimit_inv 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cocone K₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {c₂ : CategoryTheory.Limits.Cocone K₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) {c₃ : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsColimit c₃) (j : J₁ × J₂) : CategoryTheory.CategoryStruct.comp (c₃.ι.app j) (CategoryTheory.Limits.PreservesColimit₂.isoObjCoconePointsOfIsColimit G hc₁ hc₂ hc₃).inv = CategoryTheory.CategoryStruct.comp ((G.map (c₁.ι.app j.1)).app (K₂.obj j.2)) ((G.obj c₁.pt).map (c₂.ι.app j.2))
CategoryTheory.Limits.PreservesColimit₂.map_ι_comp_isoObjConePointsOfIsColimit_hom_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cocone K₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {c₂ : CategoryTheory.Limits.Cocone K₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) {c₃ : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsColimit c₃) (j : J₁ × J₂) {Z : C} (h : c₃.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp ((G.map (c₁.ι.app j.1)).app (K₂.obj j.2)) (CategoryTheory.CategoryStruct.comp ((G.obj c₁.pt).map (c₂.ι.app j.2)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesColimit₂.isoObjCoconePointsOfIsColimit G hc₁ hc₂ hc₃).hom h)) = CategoryTheory.CategoryStruct.comp (c₃.ι.app j) h
CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsLimit_hom_comp_π_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesLimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cone K₁} (hc₁ : CategoryTheory.Limits.IsLimit c₁) {c₂ : CategoryTheory.Limits.Cone K₂} (hc₂ : CategoryTheory.Limits.IsLimit c₂) {c₃ : CategoryTheory.Limits.Cone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsLimit c₃) (j : J₁ × J₂) {Z : C} (h : (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G)).obj j ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesLimit₂.isoObjConePointsOfIsLimit G hc₁ hc₂ hc₃).hom (CategoryTheory.CategoryStruct.comp (c₃.π.app j) h) = CategoryTheory.CategoryStruct.comp ((G.map (c₁.π.app j.1)).app c₂.pt) (CategoryTheory.CategoryStruct.comp ((G.obj (K₁.obj j.1)).map (c₂.π.app j.2)) h)
CategoryTheory.Limits.PreservesColimit₂.ι_comp_isoObjConePointsOfIsColimit_inv_assoc 📋 Mathlib.CategoryTheory.Limits.Preserves.Bifunctor
{J₁ : Type u_1} {J₂ : Type u_2} [CategoryTheory.Category.{v_1, u_1} J₁] [CategoryTheory.Category.{v_2, u_2} J₂] {C₁ : Type u_3} {C₂ : Type u_4} {C : Type u_5} [CategoryTheory.Category.{v_3, u_3} C₁] [CategoryTheory.Category.{v_4, u_4} C₂] [CategoryTheory.Category.{v_5, u_5} C] {K₁ : CategoryTheory.Functor J₁ C₁} {K₂ : CategoryTheory.Functor J₂ C₂} (G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C)) [CategoryTheory.Limits.PreservesColimit₂ K₁ K₂ G] {c₁ : CategoryTheory.Limits.Cocone K₁} (hc₁ : CategoryTheory.Limits.IsColimit c₁) {c₂ : CategoryTheory.Limits.Cocone K₂} (hc₂ : CategoryTheory.Limits.IsColimit c₂) {c₃ : CategoryTheory.Limits.Cocone (CategoryTheory.Functor.uncurry.obj ((((CategoryTheory.Functor.whiskeringLeft₂ C).obj K₁).obj K₂).obj G))} (hc₃ : CategoryTheory.Limits.IsColimit c₃) (j : J₁ × J₂) {Z : C} (h : (G.obj c₁.pt).obj c₂.pt ⟶ Z) : CategoryTheory.CategoryStruct.comp (c₃.ι.app j) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesColimit₂.isoObjCoconePointsOfIsColimit G hc₁ hc₂ hc₃).inv h) = CategoryTheory.CategoryStruct.comp ((G.map (c₁.ι.app j.1)).app (K₂.obj j.2)) (CategoryTheory.CategoryStruct.comp ((G.obj c₁.pt).map (c₂.ι.app j.2)) h)
CategoryTheory.IsSifted.factorization_prodComparison_colim 📋 Mathlib.CategoryTheory.Limits.Sifted
{C : Type u} [CategoryTheory.SmallCategory C] (X Y : CategoryTheory.Functor C (Type u)) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.HasColimit.isoOfNatIso ((CategoryTheory.MonoidalCategory.externalProductCompDiagIso C (Type u)).app (X, Y)).symm).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.pre (CategoryTheory.MonoidalCategory.externalProduct X Y) (CategoryTheory.Functor.diag C)) (CategoryTheory.Limits.PreservesColimit₂.isoColimitUncurryWhiskeringLeft₂ X Y (CategoryTheory.MonoidalCategory.curriedTensor (Type u))).hom) = CategoryTheory.CartesianMonoidalCategory.prodComparison CategoryTheory.Limits.colim X Y
CategoryTheory.Limits.colimitLimitToLimitColimitCone_hom 📋 Mathlib.CategoryTheory.Limits.ColimitLimit
{J : Type u₁} {K : Type u₂} [CategoryTheory.Category.{v₁, u₁} J] [CategoryTheory.Category.{v₂, u₂} K] {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasLimitsOfShape J C] [CategoryTheory.Limits.HasColimitsOfShape K C] (G : CategoryTheory.Functor J (CategoryTheory.Functor K C)) [CategoryTheory.Limits.HasLimit G] : (CategoryTheory.Limits.colimitLimitToLimitColimitCone G).hom = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colim.map (CategoryTheory.Limits.limitIsoSwapCompLim G).hom) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimitLimitToLimitColimit (CategoryTheory.Functor.uncurry.obj G)) (CategoryTheory.Limits.lim.map (CategoryTheory.Functor.whiskerRight (CategoryTheory.Functor.currying.unitIso.app G).inv CategoryTheory.Limits.colim)))
CategoryTheory.Functor.bifunctorComp₁₂Iso 📋 Mathlib.CategoryTheory.Functor.CurryingThree
{C₁ : Type u_1} {C₂ : Type u_2} {C₁₂ : Type u_3} {C₃ : Type u_4} {E : Type u_9} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_4} C₃] [CategoryTheory.Category.{v_4, u_3} C₁₂] [CategoryTheory.Category.{v_9, u_9} E] (F₁₂ : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₁₂)) (G : CategoryTheory.Functor C₁₂ (CategoryTheory.Functor C₃ E)) : CategoryTheory.bifunctorComp₁₂ F₁₂ G ≅ CategoryTheory.Functor.curry.obj ((CategoryTheory.Functor.uncurry.obj F₁₂).comp G)
CategoryTheory.Functor.bifunctorComp₂₃Iso 📋 Mathlib.CategoryTheory.Functor.CurryingThree
{C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_4} {C₂₃ : Type u_5} {E : Type u_9} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_4} C₃] [CategoryTheory.Category.{v_5, u_5} C₂₃] [CategoryTheory.Category.{v_9, u_9} E] (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂₃ E)) (G₂₃ : CategoryTheory.Functor C₂ (CategoryTheory.Functor C₃ C₂₃)) : CategoryTheory.bifunctorComp₂₃ F G₂₃ ≅ CategoryTheory.Functor.curry.obj (CategoryTheory.Functor.curry.obj ((CategoryTheory.prod.associator C₁ C₂ C₃).comp (CategoryTheory.Functor.uncurry.obj ((CategoryTheory.Functor.uncurry.obj G₂₃).comp F.flip).flip)))
CategoryTheory.Functor.bifunctorComp₁₂Iso_hom_app_app_app 📋 Mathlib.CategoryTheory.Functor.CurryingThree
{C₁ : Type u_1} {C₂ : Type u_2} {C₁₂ : Type u_3} {C₃ : Type u_4} {E : Type u_9} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_4} C₃] [CategoryTheory.Category.{v_4, u_3} C₁₂] [CategoryTheory.Category.{v_9, u_9} E] (F₁₂ : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₁₂)) (G : CategoryTheory.Functor C₁₂ (CategoryTheory.Functor C₃ E)) (X : C₁) (X✝ : C₂) (X✝¹ : C₃) : (((F₁₂.bifunctorComp₁₂Iso G).hom.app X).app X✝).app X✝¹ = CategoryTheory.CategoryStruct.id ((G.obj ((F₁₂.obj X).obj X✝)).obj X✝¹)
CategoryTheory.Functor.bifunctorComp₁₂Iso_inv_app_app_app 📋 Mathlib.CategoryTheory.Functor.CurryingThree
{C₁ : Type u_1} {C₂ : Type u_2} {C₁₂ : Type u_3} {C₃ : Type u_4} {E : Type u_9} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_4} C₃] [CategoryTheory.Category.{v_4, u_3} C₁₂] [CategoryTheory.Category.{v_9, u_9} E] (F₁₂ : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₁₂)) (G : CategoryTheory.Functor C₁₂ (CategoryTheory.Functor C₃ E)) (X : C₁) (X✝ : C₂) (X✝¹ : C₃) : (((F₁₂.bifunctorComp₁₂Iso G).inv.app X).app X✝).app X✝¹ = CategoryTheory.CategoryStruct.id ((G.obj ((F₁₂.obj X).obj X✝)).obj X✝¹)
CategoryTheory.Functor.bifunctorComp₂₃Iso_hom_app_app_app 📋 Mathlib.CategoryTheory.Functor.CurryingThree
{C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_4} {C₂₃ : Type u_5} {E : Type u_9} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_4} C₃] [CategoryTheory.Category.{v_5, u_5} C₂₃] [CategoryTheory.Category.{v_9, u_9} E] (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂₃ E)) (G₂₃ : CategoryTheory.Functor C₂ (CategoryTheory.Functor C₃ C₂₃)) (X : C₁) (X✝ : C₂) (X✝¹ : C₃) : (((F.bifunctorComp₂₃Iso G₂₃).hom.app X).app X✝).app X✝¹ = CategoryTheory.CategoryStruct.id ((F.obj X).obj ((G₂₃.obj X✝).obj X✝¹))
CategoryTheory.Functor.bifunctorComp₂₃Iso_inv_app_app_app 📋 Mathlib.CategoryTheory.Functor.CurryingThree
{C₁ : Type u_1} {C₂ : Type u_2} {C₃ : Type u_4} {C₂₃ : Type u_5} {E : Type u_9} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_4} C₃] [CategoryTheory.Category.{v_5, u_5} C₂₃] [CategoryTheory.Category.{v_9, u_9} E] (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂₃ E)) (G₂₃ : CategoryTheory.Functor C₂ (CategoryTheory.Functor C₃ C₂₃)) (X : C₁) (X✝ : C₂) (X✝¹ : C₃) : (((F.bifunctorComp₂₃Iso G₂₃).inv.app X).app X✝).app X✝¹ = CategoryTheory.CategoryStruct.id ((F.obj X).obj ((G₂₃.obj X✝).obj X✝¹))
CategoryTheory.Localization.Lifting₂.uncurry 📋 Mathlib.CategoryTheory.Localization.Bifunctor
{C₁ : Type u_1} {C₂ : Type u_2} {D₁ : Type u_3} {D₂ : Type u_4} {E : Type u_5} [CategoryTheory.Category.{v_1, u_1} C₁] [CategoryTheory.Category.{v_2, u_2} C₂] [CategoryTheory.Category.{v_3, u_3} D₁] [CategoryTheory.Category.{v_4, u_4} D₂] [CategoryTheory.Category.{v_5, u_5} E] (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) (W₁ : CategoryTheory.MorphismProperty C₁) (W₂ : CategoryTheory.MorphismProperty C₂) (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ E)) (F' : CategoryTheory.Functor D₁ (CategoryTheory.Functor D₂ E)) [CategoryTheory.Localization.Lifting₂ L₁ L₂ W₁ W₂ F F'] : CategoryTheory.Localization.Lifting (L₁.prod L₂) (W₁.prod W₂) (CategoryTheory.Functor.uncurry.obj F) (CategoryTheory.Functor.uncurry.obj F')
CategoryTheory.Functor.closedIhom_obj_map 📋 Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Groupoid
{D : Type u} {C : Type u_1} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F Y : CategoryTheory.Functor D C) {X✝ Y✝ : D} (f : X✝ ⟶ Y✝) : (F.closedIhom.obj Y).map f = CategoryTheory.CategoryStruct.comp ((CategoryTheory.MonoidalClosed.pre (CategoryTheory.inv (F.map f))).app (Y.obj X✝)) ((CategoryTheory.ihom (F.obj Y✝)).map (Y.map f))
CategoryTheory.Functor.closedIhom_map_app 📋 Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Groupoid
{D : Type u} {C : Type u_1} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F : CategoryTheory.Functor D C) {X✝ Y✝ : CategoryTheory.Functor D C} (g : X✝ ⟶ Y✝) (X : D) : (F.closedIhom.map g).app X = (CategoryTheory.ihom (F.obj X)).map (g.app X)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c