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Found 119 declarations mentioning CategoryTheory.Limits.coprod.inl.

• CategoryTheory.Limits.coprod.inl 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] : X ⟶ X ⨿ Y
• CategoryTheory.Limits.coprodIsCoprod 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr)
• CategoryTheory.Limits.coprod.desc_inl_inr 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr = CategoryTheory.CategoryStruct.id (X ⨿ Y)
• CategoryTheory.Limits.coprod.inl_desc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprod.desc f g) = f
• CategoryTheory.Limits.coprod.rightUnitor_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.Limits.HasInitial C] (P : C) : (CategoryTheory.Limits.coprod.rightUnitor P).inv = CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.coprod.inl_desc_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) {Z : C} (h : W ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc f g) h) = CategoryTheory.CategoryStruct.comp f h
• CategoryTheory.Limits.coprod.inl_map 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} [CategoryTheory.Limits.HasBinaryCoproduct W X] [CategoryTheory.Limits.HasBinaryCoproduct Y Z] (f : W ⟶ Y) (g : X ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprod.map f g) = CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.coprod.desc_comp_inl_comp_inr 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} [CategoryTheory.Limits.HasBinaryCoproduct W Y] [CategoryTheory.Limits.HasBinaryCoproduct X Z] (g : W ⟶ X) (g' : Y ⟶ Z) : CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp g CategoryTheory.Limits.coprod.inl) (CategoryTheory.CategoryStruct.comp g' CategoryTheory.Limits.coprod.inr) = CategoryTheory.Limits.coprod.map g g'
• CategoryTheory.Limits.coprod.desc' 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) : { l // CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl l = f ∧ CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr l = g }
• CategoryTheory.Limits.coprod.braiding_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q : C) : (CategoryTheory.Limits.coprod.braiding P Q).hom = CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.coprod.braiding_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q : C) : (CategoryTheory.Limits.coprod.braiding P Q).inv = CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.coprod.map_inl_inr_codiag 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] [CategoryTheory.Limits.HasBinaryCoproduct (X ⨿ Y) (X ⨿ Y)] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.map CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr) (CategoryTheory.Limits.codiag (X ⨿ Y)) = CategoryTheory.CategoryStruct.id (X ⨿ Y)
• CategoryTheory.Limits.coprod.inl_map_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y Z : C} [CategoryTheory.Limits.HasBinaryCoproduct W X] [CategoryTheory.Limits.HasBinaryCoproduct Y Z] (f : W ⟶ Y) (g : X ⟶ Z) {Z✝ : C} (h : Y ⨿ Z ⟶ Z✝) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.map f g) h) = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h)
• CategoryTheory.Limits.coprod.map_inl_inr_codiag_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] [CategoryTheory.Limits.HasBinaryCoproduct (X ⨿ Y) (X ⨿ Y)] {Z : C} (h : X ⨿ Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.map CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.codiag (X ⨿ Y)) h) = h
• CategoryTheory.Limits.coprod.hom_ext 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] {f g : X ⨿ Y ⟶ W} (h₁ : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl g) (h₂ : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr g) : f = g
• CategoryTheory.Limits.coprod.hom_ext_iff 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] {W X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] {f g : X ⨿ Y ⟶ W} : f = g ↔ CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl g ∧ CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr f = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr g
• CategoryTheory.Limits.coprod.symmetry' 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q : C) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) = CategoryTheory.CategoryStruct.id (P ⨿ Q)
• CategoryTheory.Limits.coprod.symmetry'_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q : C) {Z : C} (h : P ⨿ Q ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) h) = h
• CategoryTheory.Limits.coprod.associator_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q R : C) : (CategoryTheory.Limits.coprod.associator P Q R).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inr)
• CategoryTheory.Limits.coprod.associator_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] (P Q R : C) : (CategoryTheory.Limits.coprod.associator P Q R).inv = CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inl) (CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) CategoryTheory.Limits.coprod.inr)
• CategoryTheory.Limits.coprod.map_comp_inl_inr_codiag 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) C] {X X' Y Y' : C} (g : X ⟶ Y) (g' : X' ⟶ Y') : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.map (CategoryTheory.CategoryStruct.comp g CategoryTheory.Limits.coprod.inl) (CategoryTheory.CategoryStruct.comp g' CategoryTheory.Limits.coprod.inr)) (CategoryTheory.Limits.codiag (Y ⨿ Y')) = CategoryTheory.Limits.coprod.map g g'
• CategoryTheory.Limits.coprod.map_comp_inl_inr_codiag_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasColimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) C] {X X' Y Y' : C} (g : X ⟶ Y) (g' : X' ⟶ Y') {Z : C} (h : Y ⨿ Y' ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.map (CategoryTheory.CategoryStruct.comp g CategoryTheory.Limits.coprod.inl) (CategoryTheory.CategoryStruct.comp g' CategoryTheory.Limits.coprod.inr)) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.codiag (Y ⨿ Y')) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.map g g') h
• CategoryTheory.Limits.coprodComparison_inl 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.ProdComparison
  {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u₂} [CategoryTheory.Category.{w, u₂} D] (F : CategoryTheory.Functor C D) {A B : C} [CategoryTheory.Limits.HasBinaryCoproduct A B] [CategoryTheory.Limits.HasBinaryCoproduct (F.obj A) (F.obj B)] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprodComparison F A B) = F.map CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.coprodComparison_inl_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.ProdComparison
  {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u₂} [CategoryTheory.Category.{w, u₂} D] (F : CategoryTheory.Functor C D) {A B : C} [CategoryTheory.Limits.HasBinaryCoproduct A B] [CategoryTheory.Limits.HasBinaryCoproduct (F.obj A) (F.obj B)] {Z : D} (h : F.obj (A ⨿ B) ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprodComparison F A B) h) = CategoryTheory.CategoryStruct.comp (F.map CategoryTheory.Limits.coprod.inl) h
• CategoryTheory.Limits.map_inl_inv_coprodComparison 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.ProdComparison
  {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u₂} [CategoryTheory.Category.{w, u₂} D] (F : CategoryTheory.Functor C D) {A B : C} [CategoryTheory.Limits.HasBinaryCoproduct A B] [CategoryTheory.Limits.HasBinaryCoproduct (F.obj A) (F.obj B)] [CategoryTheory.IsIso (CategoryTheory.Limits.coprodComparison F A B)] : CategoryTheory.CategoryStruct.comp (F.map CategoryTheory.Limits.coprod.inl) (CategoryTheory.inv (CategoryTheory.Limits.coprodComparison F A B)) = CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.map_inl_inv_coprodComparison_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts.ProdComparison
  {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u₂} [CategoryTheory.Category.{w, u₂} D] (F : CategoryTheory.Functor C D) {A B : C} [CategoryTheory.Limits.HasBinaryCoproduct A B] [CategoryTheory.Limits.HasBinaryCoproduct (F.obj A) (F.obj B)] [CategoryTheory.IsIso (CategoryTheory.Limits.coprodComparison F A B)] {Z : D} (h : F.obj A ⨿ F.obj B ⟶ Z) : CategoryTheory.CategoryStruct.comp (F.map CategoryTheory.Limits.coprod.inl) (CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (CategoryTheory.Limits.coprodComparison F A B)) h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h
• CategoryTheory.Limits.isSplitMono_coprod_inl 📋 Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} [CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.pair X Y)] : CategoryTheory.IsSplitMono CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.coprod.inl_fst 📋 Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprod.fst X Y) = CategoryTheory.CategoryStruct.id X
• CategoryTheory.Limits.coprod.inl_fst_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] {Z : C} (h : X ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.fst X Y) h) = h
• CategoryTheory.Limits.coprod.inl_snd 📋 Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprod.snd X Y) = 0
• CategoryTheory.Limits.coprod.inl_snd_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] {Z : C} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.coprod.snd X Y) h) = CategoryTheory.CategoryStruct.comp 0 h
• CategoryTheory.Limits.isColimitOfHasBinaryCoproductOfPreservesColimit 📋 Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (G : CategoryTheory.Functor C D) (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] [CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.pair X Y) G] : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk (G.map CategoryTheory.Limits.coprod.inl) (G.map CategoryTheory.Limits.coprod.inr))
• CategoryTheory.Limits.biprod_isoCoprod_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
  {C : Type uC} [CategoryTheory.Category.{uC', uC} C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} [CategoryTheory.Limits.HasBinaryBiproduct X Y] : (CategoryTheory.Limits.biprod.isoCoprod X Y).hom = CategoryTheory.Limits.biprod.desc CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr
• inl_coprodIsoPushout_inv 📋 Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasPushouts C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inl (CategoryTheory.Limits.initial.to X) (CategoryTheory.Limits.initial.to Y)) (coprodIsoPushout X Y).inv = CategoryTheory.Limits.coprod.inl
• inl_coprodIsoPushout_hom 📋 Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasPushouts C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (coprodIsoPushout X Y).hom = CategoryTheory.Limits.pushout.inl (CategoryTheory.Limits.initial.to X) (CategoryTheory.Limits.initial.to Y)
• inl_coprodIsoPushout_inv_assoc 📋 Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasPushouts C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] {Z : C} (h : X ⨿ Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inl (CategoryTheory.Limits.initial.to X) (CategoryTheory.Limits.initial.to Y)) (CategoryTheory.CategoryStruct.comp (coprodIsoPushout X Y).inv h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h
• inl_coprodIsoPushout_hom_assoc 📋 Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasPushouts C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] {Z : C} (h : CategoryTheory.Limits.pushout (CategoryTheory.Limits.initial.to X) (CategoryTheory.Limits.initial.to Y) ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (coprodIsoPushout X Y).hom h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inl (CategoryTheory.Limits.initial.to X) (CategoryTheory.Limits.initial.to Y)) h
• CategoryTheory.IsPushout.of_hasBinaryCoproduct' 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] [CategoryTheory.Limits.HasInitial C] : CategoryTheory.IsPushout (CategoryTheory.Limits.initial.to X) (CategoryTheory.Limits.initial.to Y) CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr
• CategoryTheory.IsPushout.of_coprod_inl_with_id 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {A B : C} (f : A ⟶ B) (X : C) [CategoryTheory.Limits.HasBinaryCoproduct A X] [CategoryTheory.Limits.HasBinaryCoproduct B X] : CategoryTheory.IsPushout CategoryTheory.Limits.coprod.inl f (CategoryTheory.Limits.coprod.map f (CategoryTheory.CategoryStruct.id X)) CategoryTheory.Limits.coprod.inl
• CategoryTheory.algebraToUnder_obj 📋 Mathlib.CategoryTheory.Monad.Products
  {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (A : (CategoryTheory.coprodMonad X).Algebra) : (CategoryTheory.algebraToUnder X).obj A = CategoryTheory.Under.mk (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl A.a)
• CategoryTheory.coprodMonad_μ_app 📋 Mathlib.CategoryTheory.Monad.Products
  {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (x✝ : C) : (CategoryTheory.coprodMonad X).μ.app x✝ = CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.id (X ⨿ x✝))
• CategoryTheory.algebraToUnder_map 📋 Mathlib.CategoryTheory.Monad.Products
  {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] {X✝ Y✝ : (CategoryTheory.coprodMonad X).Algebra} (f : X✝ ⟶ Y✝) : (CategoryTheory.algebraToUnder X).map f = CategoryTheory.Under.homMk f.f ⋯
• CategoryTheory.Under.costar_obj_hom 📋 Mathlib.CategoryTheory.Comma.Over.Pullback
  {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) [CategoryTheory.Limits.HasBinaryCoproducts C] (X✝ : C) : ((CategoryTheory.Under.costar X).obj X✝).hom = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.id (X ⨿ X✝)))
• CategoryTheory.Limits.hasColimit_span_of_hasColimit_pair_of_hasColimit_parallelPair 📋 Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
  {C : Type u} [𝒞 : CategoryTheory.Category.{v, u} C] {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.pair Y Z)] [CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.parallelPair (CategoryTheory.CategoryStruct.comp f CategoryTheory.Limits.coprod.inl) (CategoryTheory.CategoryStruct.comp g CategoryTheory.Limits.coprod.inr))] : CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.span f g)
• CategoryTheory.Limits.inl_opProdIsoCoprod_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {A B : C} [CategoryTheory.Limits.HasBinaryProduct A B] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.opProdIsoCoprod A B).inv = CategoryTheory.Limits.prod.fst.op
• CategoryTheory.Limits.fst_opProdIsoCoprod_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {A B : C} [CategoryTheory.Limits.HasBinaryProduct A B] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst.op (CategoryTheory.Limits.opProdIsoCoprod A B).hom = CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.opProdIsoCoprod_inv_inl 📋 Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {A B : C} [CategoryTheory.Limits.HasBinaryProduct A B] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.opProdIsoCoprod A B).inv.unop CategoryTheory.Limits.coprod.inl.unop = CategoryTheory.Limits.prod.fst
• CategoryTheory.Limits.opProdIsoCoprod_hom_fst 📋 Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {A B : C} [CategoryTheory.Limits.HasBinaryProduct A B] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.opProdIsoCoprod A B).hom.unop CategoryTheory.Limits.prod.fst = CategoryTheory.Limits.coprod.inl.unop
• CategoryTheory.Limits.inl_opProdIsoCoprod_inv_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {A B : C} [CategoryTheory.Limits.HasBinaryProduct A B] {Z : Cᵒᵖ} (h : Opposite.op (A ⨯ B) ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.opProdIsoCoprod A B).inv h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst.op h
• CategoryTheory.Limits.fst_opProdIsoCoprod_hom_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {A B : C} [CategoryTheory.Limits.HasBinaryProduct A B] {Z : Cᵒᵖ} (h : Opposite.op A ⨿ Opposite.op B ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst.op (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.opProdIsoCoprod A B).hom h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h
• CategoryTheory.Limits.opProdIsoCoprod_inv_inl_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {A B : C} [CategoryTheory.Limits.HasBinaryProduct A B] {Z : C} (h : A ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.opProdIsoCoprod A B).inv.unop (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl.unop h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst h
• CategoryTheory.Limits.opProdIsoCoprod_hom_fst_assoc 📋 Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {A B : C} [CategoryTheory.Limits.HasBinaryProduct A B] {Z : C} (h : A ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.opProdIsoCoprod A B).hom.unop (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl.unop h
• CategoryTheory.Limits.Types.binaryCoproductIso_inl_comp_hom 📋 Mathlib.CategoryTheory.Limits.Types.Coproducts
  (X Y : Type u) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.Types.binaryCoproductIso X Y).hom = TypeCat.ofHom Sum.inl
• CategoryTheory.Limits.Types.binaryCoproductIso_inl_comp_inv 📋 Mathlib.CategoryTheory.Limits.Types.Coproducts
  (X Y : Type u) : CategoryTheory.CategoryStruct.comp (TypeCat.ofHom Sum.inl) (CategoryTheory.Limits.Types.binaryCoproductIso X Y).inv = CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.Types.binaryCoproductIso_inl_comp_hom_apply 📋 Mathlib.CategoryTheory.Limits.Types.Coproducts
  (X Y : Type u) (x : X) : (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.Types.binaryCoproductIso X Y).hom) ((CategoryTheory.ConcreteCategory.hom CategoryTheory.Limits.coprod.inl) x) = Sum.inl x
• CategoryTheory.Limits.Types.binaryCoproductIso_inl_comp_inv_apply 📋 Mathlib.CategoryTheory.Limits.Types.Coproducts
  (X Y : Type u) (x : X) : (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.Types.binaryCoproductIso X Y).inv) (Sum.inl x) = (CategoryTheory.ConcreteCategory.hom CategoryTheory.Limits.coprod.inl) x
• CategoryTheory.Limits.MonoCoprod.instMonoInl 📋 Mathlib.CategoryTheory.Limits.MonoCoprod
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] {A B : C} [CategoryTheory.Limits.MonoCoprod C] [CategoryTheory.Limits.HasBinaryCoproduct A B] : CategoryTheory.Mono CategoryTheory.Limits.coprod.inl
• CategoryTheory.Mono.inl_of_binaryCoproductDisjoint 📋 Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} [CategoryTheory.Limits.BinaryCoproductDisjoint X Y] [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.Mono CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.IsInitial.ofBinaryCoproductDisjoint 📋 Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} [CategoryTheory.Limits.BinaryCoproductDisjoint X Y] [CategoryTheory.Limits.HasBinaryCoproduct X Y] [CategoryTheory.Limits.HasPullback CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr] : CategoryTheory.Limits.IsInitial (CategoryTheory.Limits.pullback CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr)
• CategoryTheory.Limits.IsInitial.ofBinaryCoproductDisjointOfIsLimit 📋 Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct
  {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : C} [CategoryTheory.Limits.BinaryCoproductDisjoint X Y] [CategoryTheory.Limits.HasBinaryCoproduct X Y] (s : CategoryTheory.Limits.PullbackCone CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr) (hs : CategoryTheory.Limits.IsLimit s) : CategoryTheory.Limits.IsInitial s.pt
• CategoryTheory.HasPullbacksOfInclusions.hasPullbackInl 📋 Mathlib.CategoryTheory.Extensive
  {C : Type u} {inst✝ : CategoryTheory.Category.{v, u} C} {inst✝¹ : CategoryTheory.Limits.HasBinaryCoproducts C} [self : CategoryTheory.HasPullbacksOfInclusions C] {X Y Z : C} (f : Z ⟶ X ⨿ Y) : CategoryTheory.Limits.HasPullback CategoryTheory.Limits.coprod.inl f
• CategoryTheory.HasPullbacksOfInclusions.mk 📋 Mathlib.CategoryTheory.Extensive
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] [hasPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), CategoryTheory.Limits.HasPullback CategoryTheory.Limits.coprod.inl f] : CategoryTheory.HasPullbacksOfInclusions C
• CategoryTheory.HasPullbacksOfInclusions.preservesPullbackInl' 📋 Mathlib.CategoryTheory.Extensive
  {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.HasPullbacksOfInclusions C] {X Y Z : C} (f : Z ⟶ X ⨿ Y) : CategoryTheory.Limits.HasPullback f CategoryTheory.Limits.coprod.inl
• CategoryTheory.PreservesPullbacksOfInclusions.mk 📋 Mathlib.CategoryTheory.Extensive
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] {D : Type u_2} [CategoryTheory.Category.{v_2, u_2} D] {F : CategoryTheory.Functor C D} [CategoryTheory.Limits.HasBinaryCoproducts C] [preservesPullbackInl : ∀ {X Y Z : C} (f : Z ⟶ X ⨿ Y), CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan CategoryTheory.Limits.coprod.inl f) F] : CategoryTheory.PreservesPullbacksOfInclusions F
• CategoryTheory.PreservesPullbacksOfInclusions.preservesPullbackInl 📋 Mathlib.CategoryTheory.Extensive
  {C : Type u_1} {inst✝ : CategoryTheory.Category.{v_1, u_1} C} {D : Type u_2} {inst✝¹ : CategoryTheory.Category.{v_2, u_2} D} {F : CategoryTheory.Functor C D} {inst✝² : CategoryTheory.Limits.HasBinaryCoproducts C} [self : CategoryTheory.PreservesPullbacksOfInclusions F] {X Y Z : C} (f : Z ⟶ X ⨿ Y) : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan CategoryTheory.Limits.coprod.inl f) F
• CategoryTheory.PreservesPullbacksOfInclusions.preservesPullbackInl' 📋 Mathlib.CategoryTheory.Extensive
  {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u_1} [CategoryTheory.Category.{v_1, u_1} D] [CategoryTheory.Limits.HasBinaryCoproducts C] (F : CategoryTheory.Functor C D) [CategoryTheory.PreservesPullbacksOfInclusions F] {X Y Z : C} (f : Z ⟶ X ⨿ Y) : CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan f CategoryTheory.Limits.coprod.inl) F
• SheafOfModules.inl_freeSumIso_hom 📋 Mathlib.Algebra.Category.ModuleCat.Sheaf.Free
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrpCat] [J.WEqualsLocallyBijective AddCommGrpCat] (I J✝ : Type u) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (SheafOfModules.freeSumIso I J✝).hom = SheafOfModules.freeMap Sum.inl
• SheafOfModules.inl_freeSumIso_hom_assoc 📋 Mathlib.Algebra.Category.ModuleCat.Sheaf.Free
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {J : CategoryTheory.GrothendieckTopology C} {R : CategoryTheory.Sheaf J RingCat} [CategoryTheory.HasWeakSheafify J AddCommGrpCat] [J.WEqualsLocallyBijective AddCommGrpCat] (I J✝ : Type u) {Z : SheafOfModules R} (h : SheafOfModules.free (I ⊕ J✝) ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (SheafOfModules.freeSumIso I J✝).hom h) = CategoryTheory.CategoryStruct.comp (SheafOfModules.freeMap Sum.inl) h
• HomotopicalAlgebra.instCofibrationInlOfIsCofibrant 📋 Mathlib.AlgebraicTopology.ModelCategory.IsCofibrant
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [HomotopicalAlgebra.CategoryWithCofibrations C] [CategoryTheory.Limits.HasInitial C] (X Y : C) [(HomotopicalAlgebra.cofibrations C).IsStableUnderCobaseChange] [CategoryTheory.Limits.HasBinaryCoproduct X Y] [hY : HomotopicalAlgebra.IsCofibrant Y] : HomotopicalAlgebra.Cofibration CategoryTheory.Limits.coprod.inl
• HomotopicalAlgebra.Precylinder.inl_i 📋 Mathlib.AlgebraicTopology.ModelCategory.Cylinder
  {C : Type u} [CategoryTheory.Category.{v, u} C] {A : C} (P : HomotopicalAlgebra.Precylinder A) [CategoryTheory.Limits.HasBinaryCoproduct A A] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl P.i = P.i₀
• HomotopicalAlgebra.Precylinder.inl_i_assoc 📋 Mathlib.AlgebraicTopology.ModelCategory.Cylinder
  {C : Type u} [CategoryTheory.Category.{v, u} C] {A : C} (P : HomotopicalAlgebra.Precylinder A) [CategoryTheory.Limits.HasBinaryCoproduct A A] {Z : C} (h : P.I ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp P.i h) = CategoryTheory.CategoryStruct.comp P.i₀ h
• HomotopicalAlgebra.Cylinder.ofFactorizationData_i₀ 📋 Mathlib.AlgebraicTopology.ModelCategory.Cylinder
  {C : Type u} [CategoryTheory.Category.{v, u} C] [HomotopicalAlgebra.ModelCategory C] {A : C} (h : (HomotopicalAlgebra.cofibrations C).MapFactorizationData (HomotopicalAlgebra.trivialFibrations C) (CategoryTheory.Limits.codiag A)) : (HomotopicalAlgebra.Cylinder.ofFactorizationData h).i₀ = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h.i
• HomotopicalAlgebra.CofibrantBrownFactorization.mk'_i 📋 Mathlib.AlgebraicTopology.ModelCategory.BrownLemma
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [HomotopicalAlgebra.ModelCategory C] {X Y : C} (f : X ⟶ Y) [HomotopicalAlgebra.IsCofibrant X] [HomotopicalAlgebra.IsCofibrant Y] (h : (HomotopicalAlgebra.cofibrations C).MapFactorizationData (HomotopicalAlgebra.trivialFibrations C) (CategoryTheory.Limits.coprod.desc f (CategoryTheory.CategoryStruct.id Y))) : (HomotopicalAlgebra.CofibrantBrownFactorization.mk' f h).i = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h.i
• CategoryTheory.Limits.coprodZeroIso_inv 📋 Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] (X : C) : (CategoryTheory.Limits.coprodZeroIso X).inv = CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.inr_coprodZeroIso_hom 📋 Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] (X : C) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprodZeroIso X).hom = CategoryTheory.CategoryStruct.id X
• CategoryTheory.Limits.inl_pushoutZeroZeroIso_hom 📋 Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pushout.inl 0 0) (CategoryTheory.Limits.pushoutZeroZeroIso X Y).hom = CategoryTheory.Limits.coprod.inl
• CategoryTheory.Limits.inl_pushoutZeroZeroIso_inv 📋 Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects
  {C : Type u_1} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] (X Y : C) [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.pushoutZeroZeroIso X Y).inv = CategoryTheory.Limits.pushout.inl 0 0
• AlgebraicGeometry.instIsOpenImmersionInlScheme 📋 Mathlib.AlgebraicGeometry.Limits
  (X Y : AlgebraicGeometry.Scheme) : AlgebraicGeometry.IsOpenImmersion CategoryTheory.Limits.coprod.inl
• AlgebraicGeometry.coprodSpec_inl 📋 Mathlib.AlgebraicGeometry.Limits
  (R S : Type u) [CommRing R] [CommRing S] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (AlgebraicGeometry.coprodSpec R S) = AlgebraicGeometry.Spec.map (CommRingCat.ofHom (RingHom.fst R S))
• AlgebraicGeometry.coprodSpec_inl_assoc 📋 Mathlib.AlgebraicGeometry.Limits
  (R S : Type u) [CommRing R] [CommRing S] {Z : AlgebraicGeometry.Scheme} (h : AlgebraicGeometry.Spec (CommRingCat.of (R × S)) ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.coprodSpec R S) h) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (RingHom.fst R S))) h
• AlgebraicGeometry.isPullback_inl_inl_coprodMap 📋 Mathlib.AlgebraicGeometry.Limits
  {X Y X' Y' : AlgebraicGeometry.Scheme} (f : X ⟶ X') (g : Y ⟶ Y') : CategoryTheory.IsPullback f CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inl (CategoryTheory.Limits.coprod.map f g)
• AlgebraicGeometry.ι_left_coprodIsoSigma_inv 📋 Mathlib.AlgebraicGeometry.Limits
  (X Y : AlgebraicGeometry.Scheme) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Sigma.ι (fun i => CategoryTheory.Limits.WalkingPair.casesOn i.down X Y) { down := CategoryTheory.Limits.WalkingPair.left }) (AlgebraicGeometry.coprodIsoSigma X Y).inv = CategoryTheory.Limits.coprod.inl
• AlgebraicGeometry.isCompl_opensRange_inl_inr 📋 Mathlib.AlgebraicGeometry.Limits
  (X Y : AlgebraicGeometry.Scheme) : IsCompl (AlgebraicGeometry.Scheme.Hom.opensRange CategoryTheory.Limits.coprod.inl) (AlgebraicGeometry.Scheme.Hom.opensRange CategoryTheory.Limits.coprod.inr)
• AlgebraicGeometry.inl_ne_inr 📋 Mathlib.AlgebraicGeometry.Limits
  (X Y : AlgebraicGeometry.Scheme) (x : ↥X) (y : ↥Y) : CategoryTheory.Limits.coprod.inl x ≠ CategoryTheory.Limits.coprod.inr y
• AlgebraicGeometry.inr_ne_inl 📋 Mathlib.AlgebraicGeometry.Limits
  (X Y : AlgebraicGeometry.Scheme) (x : ↥X) (y : ↥Y) : CategoryTheory.Limits.coprod.inr y ≠ CategoryTheory.Limits.coprod.inl x
• AlgebraicGeometry.coprodMk_inl 📋 Mathlib.AlgebraicGeometry.Limits
  (X Y : AlgebraicGeometry.Scheme) (x : ↥X) : (AlgebraicGeometry.coprodMk X Y) (Sum.inl x) = CategoryTheory.Limits.coprod.inl x
• AlgebraicGeometry.isCompl_range_inl_inr 📋 Mathlib.AlgebraicGeometry.Limits
  (X Y : AlgebraicGeometry.Scheme) : IsCompl (Set.range ⇑CategoryTheory.Limits.coprod.inl) (Set.range ⇑CategoryTheory.Limits.coprod.inr)
• AlgebraicGeometry.Scheme.coprodPresheafObjIso 📋 Mathlib.AlgebraicGeometry.Limits
  {X Y : AlgebraicGeometry.Scheme} (U : (X ⨿ Y).Opens) : (X ⨿ Y).presheaf.obj (Opposite.op U) ≅ X.presheaf.obj (Opposite.op ((TopologicalSpace.Opens.map CategoryTheory.Limits.coprod.inl.base).obj U)) ⨯ Y.presheaf.obj (Opposite.op ((TopologicalSpace.Opens.map CategoryTheory.Limits.coprod.inr.base).obj U))
• AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst 📋 Mathlib.AlgebraicGeometry.Limits
  {X Y : AlgebraicGeometry.Scheme} (U : (X ⨿ Y).Opens) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.coprodPresheafObjIso U).hom CategoryTheory.Limits.prod.fst = AlgebraicGeometry.Scheme.Hom.app CategoryTheory.Limits.coprod.inl U
• AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd 📋 Mathlib.AlgebraicGeometry.Limits
  {X Y : AlgebraicGeometry.Scheme} (U : (X ⨿ Y).Opens) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.coprodPresheafObjIso U).hom CategoryTheory.Limits.prod.snd = AlgebraicGeometry.Scheme.Hom.app CategoryTheory.Limits.coprod.inr U
• AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_fst_assoc 📋 Mathlib.AlgebraicGeometry.Limits
  {X Y : AlgebraicGeometry.Scheme} (U : (X ⨿ Y).Opens) {Z : CommRingCat} (h : X.presheaf.obj (Opposite.op ((TopologicalSpace.Opens.map CategoryTheory.Limits.coprod.inl.base).obj U)) ⟶ Z) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.coprodPresheafObjIso U).hom (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.fst h) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.app CategoryTheory.Limits.coprod.inl U) h
• AlgebraicGeometry.Scheme.coprodPresheafObjIso_hom_snd_assoc 📋 Mathlib.AlgebraicGeometry.Limits
  {X Y : AlgebraicGeometry.Scheme} (U : (X ⨿ Y).Opens) {Z : CommRingCat} (h : Y.presheaf.obj (Opposite.op ((TopologicalSpace.Opens.map CategoryTheory.Limits.coprod.inr.base).obj U)) ⟶ Z) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.coprodPresheafObjIso U).hom (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd h) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.app CategoryTheory.Limits.coprod.inr U) h
• AlgebraicGeometry.Scheme.Hom.toNormalization_inl_normalizationCoprodIso_hom 📋 Mathlib.AlgebraicGeometry.Normalization
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.QuasiCompact f] [AlgebraicGeometry.QuasiSeparated f] {U V : AlgebraicGeometry.Scheme} {iU : U ⟶ X} {iV : V ⟶ X} (e : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk iU iV)) [AlgebraicGeometry.QuasiCompact iU] [AlgebraicGeometry.QuasiSeparated iU] [AlgebraicGeometry.QuasiCompact iV] [AlgebraicGeometry.QuasiSeparated iV] : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.toNormalization (CategoryTheory.CategoryStruct.comp iU f)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso f e).hom) = CategoryTheory.CategoryStruct.comp iU (AlgebraicGeometry.Scheme.Hom.toNormalization f)
• AlgebraicGeometry.Scheme.Hom.inl_normalizationCoprodIso_hom_fromNormalization 📋 Mathlib.AlgebraicGeometry.Normalization
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.QuasiCompact f] [AlgebraicGeometry.QuasiSeparated f] {U V : AlgebraicGeometry.Scheme} {iU : U ⟶ X} {iV : V ⟶ X} (e : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk iU iV)) [AlgebraicGeometry.QuasiCompact iU] [AlgebraicGeometry.QuasiSeparated iU] [AlgebraicGeometry.QuasiCompact iV] [AlgebraicGeometry.QuasiSeparated iV] : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso f e).hom (AlgebraicGeometry.Scheme.Hom.fromNormalization f)) = AlgebraicGeometry.Scheme.Hom.fromNormalization (CategoryTheory.CategoryStruct.comp iU f)
• AlgebraicGeometry.Scheme.Hom.toNormalization_inl_normalizationCoprodIso_hom_assoc 📋 Mathlib.AlgebraicGeometry.Normalization
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.QuasiCompact f] [AlgebraicGeometry.QuasiSeparated f] {U V : AlgebraicGeometry.Scheme} {iU : U ⟶ X} {iV : V ⟶ X} (e : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk iU iV)) [AlgebraicGeometry.QuasiCompact iU] [AlgebraicGeometry.QuasiSeparated iU] [AlgebraicGeometry.QuasiCompact iV] [AlgebraicGeometry.QuasiSeparated iV] {Z : AlgebraicGeometry.Scheme} (h : AlgebraicGeometry.Scheme.Hom.normalization f ⟶ Z) : CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.toNormalization (CategoryTheory.CategoryStruct.comp iU f)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso f e).hom h)) = CategoryTheory.CategoryStruct.comp iU (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.toNormalization f) h)
• AlgebraicGeometry.Scheme.Hom.inl_normalizationCoprodIso_hom_fromNormalization_assoc 📋 Mathlib.AlgebraicGeometry.Normalization
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.QuasiCompact f] [AlgebraicGeometry.QuasiSeparated f] {U V : AlgebraicGeometry.Scheme} {iU : U ⟶ X} {iV : V ⟶ X} (e : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk iU iV)) [AlgebraicGeometry.QuasiCompact iU] [AlgebraicGeometry.QuasiSeparated iU] [AlgebraicGeometry.QuasiCompact iV] [AlgebraicGeometry.QuasiSeparated iV] {Z : AlgebraicGeometry.Scheme} (h : Y ⟶ Z) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso f e).hom (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.fromNormalization f) h)) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.fromNormalization (CategoryTheory.CategoryStruct.comp iU f)) h
• AlgebraicGeometry.Scheme.Hom.inl_toNormalization_normalizationCoprodIso_inv_assoc 📋 Mathlib.AlgebraicGeometry.Normalization
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.QuasiCompact f] [AlgebraicGeometry.QuasiSeparated f] {U V : AlgebraicGeometry.Scheme} {iU : U ⟶ X} {iV : V ⟶ X} (e : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk iU iV)) [AlgebraicGeometry.QuasiCompact iU] [AlgebraicGeometry.QuasiSeparated iU] [AlgebraicGeometry.QuasiCompact iV] [AlgebraicGeometry.QuasiSeparated iV] {Z : AlgebraicGeometry.Scheme} (h : AlgebraicGeometry.Scheme.Hom.normalization (CategoryTheory.CategoryStruct.comp iU f) ⨿ AlgebraicGeometry.Scheme.Hom.normalization (CategoryTheory.CategoryStruct.comp iV f) ⟶ Z) : CategoryTheory.CategoryStruct.comp iU (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.toNormalization f) (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso f e).inv h)) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.toNormalization (CategoryTheory.CategoryStruct.comp iU f)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h)
• AlgebraicGeometry.Scheme.Hom.inl_toNormalization_normalizationCoprodIso_inv 📋 Mathlib.AlgebraicGeometry.Normalization
  {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.QuasiCompact f] [AlgebraicGeometry.QuasiSeparated f] {U V : AlgebraicGeometry.Scheme} {iU : U ⟶ X} {iV : V ⟶ X} (e : CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk iU iV)) [AlgebraicGeometry.QuasiCompact iU] [AlgebraicGeometry.QuasiSeparated iU] [AlgebraicGeometry.QuasiCompact iV] [AlgebraicGeometry.QuasiSeparated iV] : CategoryTheory.CategoryStruct.comp iU (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.toNormalization f) (AlgebraicGeometry.Scheme.Hom.normalizationCoprodIso f e).inv) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.toNormalization (CategoryTheory.CategoryStruct.comp iU f)) CategoryTheory.Limits.coprod.inl
• CategoryTheory.coprod_inl_leftDistrib_hom 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalLeftDistrib C] {X Y Z : C} : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.leftDistrib X Y Z).hom = CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inl
• CategoryTheory.coprod_inl_rightDistrib_hom 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalRightDistrib C] {X Y Z : C} : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (∂R X Y Z).hom = CategoryTheory.MonoidalCategoryStruct.whiskerRight CategoryTheory.Limits.coprod.inl X
• CategoryTheory.leftDistrib_hom 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalLeftDistrib C] {X Y Z : C} : (CategoryTheory.leftDistrib X Y Z).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inl) (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inr)
• CategoryTheory.rightDistrib_hom 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalRightDistrib C] {X Y Z : C} : (∂R X Y Z).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.MonoidalCategoryStruct.whiskerRight CategoryTheory.Limits.coprod.inl X) (CategoryTheory.MonoidalCategoryStruct.whiskerRight CategoryTheory.Limits.coprod.inr X)
• CategoryTheory.whiskerLeft_coprod_inl_leftDistrib_inv 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalLeftDistrib C] {X Y Z : C} : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inl) (CategoryTheory.leftDistrib X Y Z).inv = CategoryTheory.Limits.coprod.inl
• CategoryTheory.whiskerRight_coprod_inl_rightDistrib_inv 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalRightDistrib C] {X Y Z : C} : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight CategoryTheory.Limits.coprod.inl X) (∂R X Y Z).inv = CategoryTheory.Limits.coprod.inl
• CategoryTheory.coprod_inl_leftDistrib_hom_assoc 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalLeftDistrib C] {X Y Z Z✝ : C} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj X (Y ⨿ Z) ⟶ Z✝) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (CategoryTheory.leftDistrib X Y Z).hom h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inl) h
• CategoryTheory.coprod_inl_rightDistrib_hom_assoc 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalRightDistrib C] {X Y Z Z✝ : C} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj (Y ⨿ Z) X ⟶ Z✝) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp (∂R X Y Z).hom h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight CategoryTheory.Limits.coprod.inl X) h
• CategoryTheory.whiskerLeft_coprod_inl_leftDistrib_inv_assoc 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalLeftDistrib C] {X Y Z Z✝ : C} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj X Y ⨿ CategoryTheory.MonoidalCategoryStruct.tensorObj X Z ⟶ Z✝) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerLeft X CategoryTheory.Limits.coprod.inl) (CategoryTheory.CategoryStruct.comp (CategoryTheory.leftDistrib X Y Z).inv h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h
• CategoryTheory.whiskerRight_coprod_inl_rightDistrib_inv_assoc 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.IsMonoidalRightDistrib C] {X Y Z Z✝ : C} (h : CategoryTheory.MonoidalCategoryStruct.tensorObj Y X ⨿ CategoryTheory.MonoidalCategoryStruct.tensorObj Z X ⟶ Z✝) : CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategoryStruct.whiskerRight CategoryTheory.Limits.coprod.inl X) (CategoryTheory.CategoryStruct.comp (∂R X Y Z).inv h) = CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl h
• CategoryTheory.MonoidalClosed.leftDistrib_inv 📋 Mathlib.CategoryTheory.Distributive.Monoidal
  {C : Type u_1} [CategoryTheory.Category.{v, u_1} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.Limits.HasBinaryCoproducts C] [CategoryTheory.MonoidalClosed C] {X Y Z : C} : (CategoryTheory.leftDistrib X Y Z).inv = CategoryTheory.MonoidalClosed.uncurry (CategoryTheory.Limits.coprod.desc (CategoryTheory.MonoidalClosed.curry CategoryTheory.Limits.coprod.inl) (CategoryTheory.MonoidalClosed.curry CategoryTheory.Limits.coprod.inr))
• CategoryTheory.FunctorToTypes.inl_comp_binaryCoproductIso_hom 📋 Mathlib.CategoryTheory.Limits.Shapes.FunctorToTypes
  {C : Type u} [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C (Type w)) : CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl (CategoryTheory.FunctorToTypes.binaryCoproductIso F G).hom = CategoryTheory.FunctorToTypes.coprod.inl
• CategoryTheory.FunctorToTypes.inl_comp_binaryCoproductIso_inv 📋 Mathlib.CategoryTheory.Limits.Shapes.FunctorToTypes
  {C : Type u} [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C (Type w)) : CategoryTheory.CategoryStruct.comp CategoryTheory.FunctorToTypes.coprod.inl (CategoryTheory.FunctorToTypes.binaryCoproductIso F G).inv = CategoryTheory.Limits.coprod.inl
• CategoryTheory.FunctorToTypes.inl_comp_binaryCoproductIso_hom_apply 📋 Mathlib.CategoryTheory.Limits.Shapes.FunctorToTypes
  {C : Type u} [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C (Type w)) (a : C) (x : F.obj a) : (CategoryTheory.ConcreteCategory.hom ((CategoryTheory.FunctorToTypes.binaryCoproductIso F G).hom.app a)) ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.coprod.inl.app a)) x) = Sum.inl x
• CategoryTheory.FunctorToTypes.inl_comp_binaryCoproductIso_inv_apply 📋 Mathlib.CategoryTheory.Limits.Shapes.FunctorToTypes
  {C : Type u} [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C (Type w)) (a : C) (x : F.obj a) : (CategoryTheory.ConcreteCategory.hom ((CategoryTheory.FunctorToTypes.binaryCoproductIso F G).inv.app a)) (Sum.inl x) = (CategoryTheory.ConcreteCategory.hom (CategoryTheory.Limits.coprod.inl.app a)) x
• CategoryTheory.IsPushout.of_hasBinaryCoproduct 📋 Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C} [CategoryTheory.Limits.HasBinaryCoproduct X Y] : CategoryTheory.IsPushout 0 0 CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr
• CategoryTheory.monoidalOfHasFiniteCoproducts.rightUnitor_inv 📋 Mathlib.CategoryTheory.Monoidal.OfHasFiniteProducts
  (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasBinaryCoproducts C] (X : C) : (CategoryTheory.MonoidalCategoryStruct.rightUnitor X).inv = CategoryTheory.Limits.coprod.inl
• CategoryTheory.monoidalOfHasFiniteCoproducts.associator_hom 📋 Mathlib.CategoryTheory.Monoidal.OfHasFiniteProducts
  (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasBinaryCoproducts C] (X Y Z : C) : (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).hom = CategoryTheory.Limits.coprod.desc (CategoryTheory.Limits.coprod.desc CategoryTheory.Limits.coprod.inl (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inr)) (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inr)
• CategoryTheory.monoidalOfHasFiniteCoproducts.associator_inv 📋 Mathlib.CategoryTheory.Monoidal.OfHasFiniteProducts
  (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.HasInitial C] [CategoryTheory.Limits.HasBinaryCoproducts C] (X Y Z : C) : (CategoryTheory.MonoidalCategoryStruct.associator X Y Z).inv = CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inl CategoryTheory.Limits.coprod.inl) (CategoryTheory.Limits.coprod.desc (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr CategoryTheory.Limits.coprod.inl) CategoryTheory.Limits.coprod.inr)
• CompHausLike.HasExplicitPullbacksOfInclusions.hasProp 📋 Mathlib.Topology.Category.CompHausLike.Limits
  {P : TopCat → Prop} {inst✝ : CompHausLike.HasExplicitFiniteCoproducts P} [self : CompHausLike.HasExplicitPullbacksOfInclusions P] {X Y Z : CompHausLike P} (f : Z ⟶ X ⨿ Y) : CompHausLike.HasExplicitPullback CategoryTheory.Limits.coprod.inl f
• CompHausLike.HasExplicitPullbacksOfInclusions.mk 📋 Mathlib.Topology.Category.CompHausLike.Limits
  {P : TopCat → Prop} [CompHausLike.HasExplicitFiniteCoproducts P] (hasProp : ∀ {X Y Z : CompHausLike P} (f : Z ⟶ X ⨿ Y), CompHausLike.HasExplicitPullback CategoryTheory.Limits.coprod.inl f) : CompHausLike.HasExplicitPullbacksOfInclusions P

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:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. 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

4. 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.

5. 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