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Found 117 declarations mentioning CategoryTheory.Functor.LeftExtension.

• CategoryTheory.Functor.LeftExtension 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) : Type (max (max (max (max u_3 u_5) v_3) v_5) u_1 v_3)
• CategoryTheory.Functor.LeftExtension.mk 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (F' : CategoryTheory.Functor D H) {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (α : F ⟶ L.comp F') : L.LeftExtension F
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) : L.LeftExtension F ≌ L'.LeftExtension F
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₂ 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) {F F' : CategoryTheory.Functor C H} (iso₂ : F ≅ F') : L.LeftExtension F ≌ L.LeftExtension F'
• CategoryTheory.Functor.LeftExtension.postcompose₂ 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor H D') : CategoryTheory.Functor (L.LeftExtension F) (L.LeftExtension (F.comp G))
• CategoryTheory.Functor.LeftExtension.precomp 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {C' : Type u_2} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} C'] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor C' C) : CategoryTheory.Functor (L.LeftExtension F) ((G.comp L).LeftExtension (G.comp F))
• CategoryTheory.Functor.instIsEquivalenceLeftExtensionCompPostcompose₂ 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor H D') [G.IsEquivalence] : (CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).IsEquivalence
• CategoryTheory.Functor.LeftExtension.postcomp₁ 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D'} (G : CategoryTheory.Functor D D') (f : L' ⟶ L.comp G) (F : CategoryTheory.Functor C H) : CategoryTheory.Functor (L'.LeftExtension F) (L.LeftExtension F)
• CategoryTheory.Functor.instIsEquivalenceLeftExtensionCompPrecomp 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {C' : Type u_2} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} C'] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor C' C) [G.IsEquivalence] : (CategoryTheory.Functor.LeftExtension.precomp L F G).IsEquivalence
• CategoryTheory.Functor.LeftExtension.precomp₂ 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {F₀ : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} {F₁ : CategoryTheory.Functor D H} (L' : CategoryTheory.Functor D D') (α : F₀ ⟶ L.comp F₁) : CategoryTheory.Functor (L'.LeftExtension F₁) ((L.comp L').LeftExtension F₀)
• CategoryTheory.Functor.instIsEquivalenceLeftExtensionPostcomp₁OfIsIso 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D'} (G : CategoryTheory.Functor D D') [G.IsEquivalence] (f : L' ⟶ L.comp G) [CategoryTheory.IsIso f] (F : CategoryTheory.Functor C H) : (CategoryTheory.Functor.LeftExtension.postcomp₁ G f F).IsEquivalence
• CategoryTheory.Functor.LeftExtension.isUniversalPrecompEquiv 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {C' : Type u_2} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} C'] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor C' C) [G.IsEquivalence] (e : L.LeftExtension F) : CategoryTheory.StructuredArrow.IsUniversal e ≃ CategoryTheory.StructuredArrow.IsUniversal ((CategoryTheory.Functor.LeftExtension.precomp L F G).obj e)
• CategoryTheory.Functor.LeftExtension.isUniversalPostcomp₁Equiv 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D'} (G : CategoryTheory.Functor D D') [G.IsEquivalence] (e : L.comp G ≅ L') (F : CategoryTheory.Functor C H) (ex : L'.LeftExtension F) : CategoryTheory.StructuredArrow.IsUniversal ex ≃ CategoryTheory.StructuredArrow.IsUniversal ((CategoryTheory.Functor.LeftExtension.postcomp₁ G e.inv F).obj ex)
• CategoryTheory.Functor.LeftExtension.precomp₂_obj_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {F₀ : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} {F₁ : CategoryTheory.Functor D H} (L' : CategoryTheory.Functor D D') (α : F₀ ⟶ L.comp F₁) (X : L'.LeftExtension F₁) : ((CategoryTheory.Functor.LeftExtension.precomp₂ L' α).obj X).left = X.left
• CategoryTheory.Functor.LeftExtension.precomp₂_obj_right 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {F₀ : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} {F₁ : CategoryTheory.Functor D H} (L' : CategoryTheory.Functor D D') (α : F₀ ⟶ L.comp F₁) (X : L'.LeftExtension F₁) : ((CategoryTheory.Functor.LeftExtension.precomp₂ L' α).obj X).right = X.right
• CategoryTheory.Functor.LeftExtension.postcompose₂_obj_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor H D') (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj X).left = X.left
• CategoryTheory.Functor.LeftExtension.isUniversalOfPrecomp₂ 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor D D'} {F₀ : CategoryTheory.Functor C H} {F₁ : CategoryTheory.Functor D H} (α : F₀ ⟶ L.comp F₁) (hα : CategoryTheory.StructuredArrow.IsUniversal (CategoryTheory.Functor.LeftExtension.mk F₁ α)) {b : L'.LeftExtension F₁} (hb : CategoryTheory.StructuredArrow.IsUniversal ((CategoryTheory.Functor.LeftExtension.precomp₂ L' α).obj b)) : CategoryTheory.StructuredArrow.IsUniversal b
• CategoryTheory.Functor.LeftExtension.isUniversalPrecomp₂ 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor D D'} {F₀ : CategoryTheory.Functor C H} {F₁ : CategoryTheory.Functor D H} (α : F₀ ⟶ L.comp F₁) (hα : CategoryTheory.StructuredArrow.IsUniversal (CategoryTheory.Functor.LeftExtension.mk F₁ α)) {b : L'.LeftExtension F₁} (hb : CategoryTheory.StructuredArrow.IsUniversal b) : CategoryTheory.StructuredArrow.IsUniversal ((CategoryTheory.Functor.LeftExtension.precomp₂ L' α).obj b)
• CategoryTheory.Functor.LeftExtension.isUniversalPrecomp₂Equiv 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor D D'} {F₀ : CategoryTheory.Functor C H} {F₁ : CategoryTheory.Functor D H} (α : F₀ ⟶ L.comp F₁) (hα : CategoryTheory.StructuredArrow.IsUniversal (CategoryTheory.Functor.LeftExtension.mk F₁ α)) (b : L'.LeftExtension F₁) : CategoryTheory.StructuredArrow.IsUniversal b ≃ CategoryTheory.StructuredArrow.IsUniversal ((CategoryTheory.Functor.LeftExtension.precomp₂ L' α).obj b)
• CategoryTheory.Functor.LeftExtension.postcompose₂ObjMkIso 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (G : CategoryTheory.Functor H D') {F' : CategoryTheory.Functor D H} (α : F ⟶ L.comp F') : (CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj (CategoryTheory.Functor.LeftExtension.mk F' α) ≅ CategoryTheory.Functor.LeftExtension.mk (F'.comp G) (CategoryTheory.CategoryStruct.comp (CategoryTheory.Functor.whiskerRight α G) (L.associator F' G).hom)
• CategoryTheory.Functor.LeftExtension.postcomp₁_obj_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D'} (G : CategoryTheory.Functor D D') (f : L' ⟶ L.comp G) (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D' H).obj L')) : ((CategoryTheory.Functor.LeftExtension.postcomp₁ G f F).obj X).left = X.left
• CategoryTheory.Functor.LeftExtension.postcompose₂_obj_right_obj 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor H D') (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) (X✝ : D) : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj X).right.obj X✝ = G.obj (X.right.obj X✝)
• CategoryTheory.Functor.LeftExtension.precomp_obj_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {C' : Type u_2} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} C'] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor C' C) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) : ((CategoryTheory.Functor.LeftExtension.precomp L F G).obj X).left = X.left
• CategoryTheory.Functor.LeftExtension.precomp_obj_right 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {C' : Type u_2} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} C'] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor C' C) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) : ((CategoryTheory.Functor.LeftExtension.precomp L F G).obj X).right = X.right
• CategoryTheory.Functor.LeftExtension.postcomp₁_obj_right_obj 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D'} (G : CategoryTheory.Functor D D') (f : L' ⟶ L.comp G) (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D' H).obj L')) (X✝ : D) : ((CategoryTheory.Functor.LeftExtension.postcomp₁ G f F).obj X).right.obj X✝ = X.right.obj (G.obj X✝)
• CategoryTheory.Functor.LeftExtension.postcomp₁_obj_right_map 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D'} (G : CategoryTheory.Functor D D') (f : L' ⟶ L.comp G) (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D' H).obj L')) {X✝ Y✝ : D} (f✝ : X✝ ⟶ Y✝) : ((CategoryTheory.Functor.LeftExtension.postcomp₁ G f F).obj X).right.map f✝ = X.right.map (G.map f✝)
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_functor_obj_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).functor.obj X).left = X.left
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_inverse_obj_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L')) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).inverse.obj X).left = X.left
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_functor_obj_right 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).functor.obj X).right = X.right
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_inverse_obj_right 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L')) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).inverse.obj X).right = X.right
• CategoryTheory.Functor.LeftExtension.postcompose₂_obj_right_map 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor H D') (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) {X✝ Y✝ : D} (f : X✝ ⟶ Y✝) : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj X).right.map f = G.map (X.right.map f)
• CategoryTheory.Functor.LeftExtension.precomp_obj_hom_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {C' : Type u_2} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} C'] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor C' C) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) (X✝ : C') : ((CategoryTheory.Functor.LeftExtension.precomp L F G).obj X).hom.app X✝ = X.hom.app (G.obj X✝)
• CategoryTheory.Functor.LeftExtension.isUniversalEquivOfIso₂ 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L : CategoryTheory.Functor C D} {F₁ F₂ : CategoryTheory.Functor C H} (α₁ : L.LeftExtension F₁) (α₂ : L.LeftExtension F₂) (e : F₁ ≅ F₂) (e' : CategoryTheory.StructuredArrow.right α₁ ≅ CategoryTheory.StructuredArrow.right α₂) (h : CategoryTheory.CategoryStruct.comp (CategoryTheory.StructuredArrow.hom α₁) (L.whiskerLeft e'.hom) = CategoryTheory.CategoryStruct.comp e.hom (CategoryTheory.StructuredArrow.hom α₂)) : CategoryTheory.StructuredArrow.IsUniversal α₁ ≃ CategoryTheory.StructuredArrow.IsUniversal α₂
• CategoryTheory.Functor.LeftExtension.postcompose₂_obj_hom_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor H D') (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) (X✝ : C) : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj X).hom.app X✝ = G.map (X.hom.app X✝)
• CategoryTheory.Functor.LeftExtension.postcomp₁_obj_hom_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D'} (G : CategoryTheory.Functor D D') (f : L' ⟶ L.comp G) (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D' H).obj L')) (X✝ : C) : ((CategoryTheory.Functor.LeftExtension.postcomp₁ G f F).obj X).hom.app X✝ = CategoryTheory.CategoryStruct.comp (X.hom.app X✝) (X.right.map (f.app X✝))
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_functor_obj_hom_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) (X✝ : C) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).functor.obj X).hom.app X✝ = CategoryTheory.CategoryStruct.comp (X.hom.app X✝) (X.right.map (iso₁.hom.app X✝))
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_inverse_obj_hom_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L')) (X✝ : C) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).inverse.obj X).hom.app X✝ = CategoryTheory.CategoryStruct.comp (X.hom.app X✝) (X.right.map (iso₁.inv.app X✝))
• CategoryTheory.Functor.LeftExtension.postcompose₂ObjMkIso_hom_right_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (G : CategoryTheory.Functor H D') {F' : CategoryTheory.Functor D H} (α : F ⟶ L.comp F') (X : D) : (CategoryTheory.Functor.LeftExtension.postcompose₂ObjMkIso G α).hom.right.app X = CategoryTheory.CategoryStruct.id (G.obj (F'.obj X))
• CategoryTheory.Functor.LeftExtension.postcompose₂ObjMkIso_inv_right_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (G : CategoryTheory.Functor H D') {F' : CategoryTheory.Functor D H} (α : F ⟶ L.comp F') (X : D) : (CategoryTheory.Functor.LeftExtension.postcompose₂ObjMkIso G α).inv.right.app X = CategoryTheory.CategoryStruct.id (G.obj (F'.obj X))
• CategoryTheory.Functor.LeftExtension.precomp₂_obj_hom_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {F₀ : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} {F₁ : CategoryTheory.Functor D H} (L' : CategoryTheory.Functor D D') (α : F₀ ⟶ L.comp F₁) (X : L'.LeftExtension F₁) (X✝ : C) : ((CategoryTheory.Functor.LeftExtension.precomp₂ L' α).obj X).hom.app X✝ = CategoryTheory.CategoryStruct.comp (α.app X✝) (X.hom.app (L.obj X✝))
• CategoryTheory.Functor.LeftExtension.precomp₂_map_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {F₀ : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} {F₁ : CategoryTheory.Functor D H} (L' : CategoryTheory.Functor D D') (α : F₀ ⟶ L.comp F₁) {X✝ Y✝ : L'.LeftExtension F₁} (f : X✝ ⟶ Y✝) : ((CategoryTheory.Functor.LeftExtension.precomp₂ L' α).map f).left = CategoryTheory.CategoryStruct.id X✝.left
• CategoryTheory.Functor.LeftExtension.precomp₂_map_right 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {F₀ : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} {F₁ : CategoryTheory.Functor D H} (L' : CategoryTheory.Functor D D') (α : F₀ ⟶ L.comp F₁) {X✝ Y✝ : L'.LeftExtension F₁} (f : X✝ ⟶ Y✝) : ((CategoryTheory.Functor.LeftExtension.precomp₂ L' α).map f).right = f.right
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_functor_map_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) {Y✝ X✝ : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)} (f : Y✝ ⟶ X✝) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).functor.map f).left = CategoryTheory.CategoryStruct.id Y✝.left
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_inverse_map_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) {Y✝ X✝ : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L')} (f : Y✝ ⟶ X✝) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).inverse.map f).left = CategoryTheory.CategoryStruct.id Y✝.left
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_functor_map_right 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) {Y✝ X✝ : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)} (f : Y✝ ⟶ X✝) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).functor.map f).right = f.right
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_inverse_map_right 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) {Y✝ X✝ : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L')} (f : Y✝ ⟶ X✝) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).inverse.map f).right = f.right
• CategoryTheory.Functor.LeftExtension.postcomp₁_map_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D'} (G : CategoryTheory.Functor D D') (f : L' ⟶ L.comp G) (F : CategoryTheory.Functor C H) {X Y : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D' H).obj L')} (φ : X ⟶ Y) : ((CategoryTheory.Functor.LeftExtension.postcomp₁ G f F).map φ).left = CategoryTheory.CategoryStruct.id X.left
• CategoryTheory.Functor.LeftExtension.postcomp₁_map_right_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] {L : CategoryTheory.Functor C D} {L' : CategoryTheory.Functor C D'} (G : CategoryTheory.Functor D D') (f : L' ⟶ L.comp G) (F : CategoryTheory.Functor C H) {X Y : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D' H).obj L')} (φ : X ⟶ Y) (X✝ : D) : ((CategoryTheory.Functor.LeftExtension.postcomp₁ G f F).map φ).right.app X✝ = φ.right.app (G.obj X✝)
• CategoryTheory.Functor.LeftExtension.precomp_map_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {C' : Type u_2} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} C'] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor C' C) {X Y : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)} (φ : X ⟶ Y) : ((CategoryTheory.Functor.LeftExtension.precomp L F G).map φ).left = CategoryTheory.CategoryStruct.id X.left
• CategoryTheory.Functor.LeftExtension.precomp_map_right 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {C' : Type u_2} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} C'] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor C' C) {X Y : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)} (φ : X ⟶ Y) : ((CategoryTheory.Functor.LeftExtension.precomp L F G).map φ).right = φ.right
• CategoryTheory.Functor.LeftExtension.postcompose₂_map_left 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor H D') {X Y : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)} (φ : X ⟶ Y) : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).map φ).left = CategoryTheory.CategoryStruct.id X.left
• CategoryTheory.Functor.LeftExtension.postcompose₂_map_right_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} {D' : Type u_6} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] [CategoryTheory.Category.{v_6, u_6} D'] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (G : CategoryTheory.Functor H D') {X Y : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)} (φ : X ⟶ Y) (X✝ : D) : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).map φ).right.app X✝ = G.map (φ.right.app X✝)
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_counitIso_hom_app_right_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L')) (X✝ : D) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).counitIso.hom.app X).right.app X✝ = CategoryTheory.CategoryStruct.id (X.right.obj X✝)
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_counitIso_inv_app_right_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L')) (X✝ : D) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).counitIso.inv.app X).right.app X✝ = CategoryTheory.CategoryStruct.id (X.right.obj X✝)
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_unitIso_hom_app_right_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) (X✝ : D) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).unitIso.hom.app X).right.app X✝ = CategoryTheory.CategoryStruct.id (X.right.obj X✝)
• CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁_unitIso_inv_app_right_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Basic
  {C : Type u_1} {H : Type u_3} {D : Type u_5} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_3, u_3} H] [CategoryTheory.Category.{v_5, u_5} D] {L L' : CategoryTheory.Functor C D} (iso₁ : L ≅ L') (F : CategoryTheory.Functor C H) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.Functor.whiskeringLeft C D H).obj L)) (X✝ : D) : ((CategoryTheory.Functor.leftExtensionEquivalenceOfIso₁ iso₁ F).unitIso.inv.app X).right.app X✝ = CategoryTheory.CategoryStruct.id (X.right.obj X✝)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) : Type (max (max (max (max u_2 u_4) v_4) v_2) u_1)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) (Y : D) : Type (max (max (max u_1 v_2) u_4) v_4)
• CategoryTheory.Functor.LeftExtension.isPointwiseLeftKanExtensionAt 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) : CategoryTheory.ObjectProperty D
• CategoryTheory.Functor.LeftExtension.instIsClosedUnderIsomorphismsIsPointwiseLeftKanExtensionAt 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) : E.isPointwiseLeftKanExtensionAt.IsClosedUnderIsomorphisms
• CategoryTheory.Functor.LeftExtension.instSubsingletonIsPointwiseLeftKanExtensionAt 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) (Y : D) : Subsingleton (E.IsPointwiseLeftKanExtensionAt Y)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.hasLeftKanExtension 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) : L.HasLeftKanExtension F
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.hasPointwiseLeftKanExtension 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) : L.HasPointwiseLeftKanExtension F
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.hasPointwiseLeftKanExtensionAt 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) : L.HasPointwiseLeftKanExtensionAt F Y
• CategoryTheory.Functor.LeftExtension.isPointwiseLeftKanExtensionAtOfIso' 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) {Y : D} (hY : E.IsPointwiseLeftKanExtensionAt Y) {Y' : D} (e : Y ≅ Y') : E.IsPointwiseLeftKanExtensionAt Y'
• CategoryTheory.Functor.LeftExtension.isPointwiseLeftKanExtensionAtEquivOfIso' 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) {Y Y' : D} (e : Y ≅ Y') : E.IsPointwiseLeftKanExtensionAt Y ≃ E.IsPointwiseLeftKanExtensionAt Y'
• CategoryTheory.Functor.LeftExtension.coconeAt 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) (Y : D) : CategoryTheory.Limits.Cocone ((CategoryTheory.CostructuredArrow.proj L Y).comp F)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.isUniversal 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) : CategoryTheory.StructuredArrow.IsUniversal E
• CategoryTheory.Functor.LeftExtension.isPointwiseLeftKanExtensionEquivOfIso 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E E' : L.LeftExtension F} (e : E ≅ E') : E.IsPointwiseLeftKanExtension ≃ E'.IsPointwiseLeftKanExtension
• CategoryTheory.Functor.LeftExtension.isPointwiseLeftKanExtensionAtEquivOfIso 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E E' : L.LeftExtension F} (e : E ≅ E') (Y : D) : E.IsPointwiseLeftKanExtensionAt Y ≃ E'.IsPointwiseLeftKanExtensionAt Y
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.homFrom 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) (G : L.LeftExtension F) : E ⟶ G
• CategoryTheory.Functor.LeftExtension.coconeAt_pt 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) (Y : D) : (E.coconeAt Y).pt = (CategoryTheory.StructuredArrow.right E).obj Y
• CategoryTheory.Functor.LeftExtension.coconeAtFunctor 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (Y : D) : CategoryTheory.Functor (L.LeftExtension F) (CategoryTheory.Limits.Cocone ((CategoryTheory.CostructuredArrow.proj L Y).comp F))
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.isLeftKanExtension 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) : (CategoryTheory.StructuredArrow.right E).IsLeftKanExtension (CategoryTheory.StructuredArrow.hom E)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.isoColimit 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) [CategoryTheory.Limits.HasColimit ((CategoryTheory.CostructuredArrow.proj L Y).comp F)] : (CategoryTheory.StructuredArrow.right E).obj Y ≅ CategoryTheory.Limits.colimit ((CategoryTheory.CostructuredArrow.proj L Y).comp F)
• CategoryTheory.Functor.LeftExtension.coconeAtFunctor_obj 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (Y : D) (E : L.LeftExtension F) : (CategoryTheory.Functor.LeftExtension.coconeAtFunctor L F Y).obj E = E.coconeAt Y
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.isIso_hom 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) [L.Full] [L.Faithful] : CategoryTheory.IsIso (CategoryTheory.StructuredArrow.hom E)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.hom_ext 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) {G : L.LeftExtension F} {f₁ f₂ : E ⟶ G} : f₁ = f₂
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.isIso_hom_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) {X : C} (h : E.IsPointwiseLeftKanExtensionAt (L.obj X)) [L.Full] [L.Faithful] : CategoryTheory.IsIso ((CategoryTheory.StructuredArrow.hom E).app X)
• CategoryTheory.Functor.LeftExtension.coconeAtFunctor_map_hom 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] (L : CategoryTheory.Functor C D) (F : CategoryTheory.Functor C H) (Y : D) {E E' : L.LeftExtension F} (φ : E ⟶ E') : ((CategoryTheory.Functor.LeftExtension.coconeAtFunctor L F Y).map φ).hom = (CategoryTheory.StructuredArrow.Hom.right φ).app Y
• CategoryTheory.Functor.LeftExtension.coconeAt_ι_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} (E : L.LeftExtension F) (Y : D) (g : CategoryTheory.CostructuredArrow L Y) : (E.coconeAt Y).ι.app g = CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app g.left) ((CategoryTheory.StructuredArrow.right E).map g.hom)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.ι_isoColimit_inv 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) [CategoryTheory.Limits.HasColimit ((CategoryTheory.CostructuredArrow.proj L Y).comp F)] (g : CategoryTheory.CostructuredArrow L Y) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.CostructuredArrow.proj L Y).comp F) g) h.isoColimit.inv = CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app g.left) ((CategoryTheory.StructuredArrow.right E).map g.hom)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.ι_isoColimit_hom 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) [CategoryTheory.Limits.HasColimit ((CategoryTheory.CostructuredArrow.proj L Y).comp F)] (g : CategoryTheory.CostructuredArrow L Y) : CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app g.left) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.right E).map g.hom) h.isoColimit.hom) = CategoryTheory.Limits.colimit.ι ((CategoryTheory.CostructuredArrow.proj L Y).comp F) g
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.ι_isoColimit_hom_assoc 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) [CategoryTheory.Limits.HasColimit ((CategoryTheory.CostructuredArrow.proj L Y).comp F)] (g : CategoryTheory.CostructuredArrow L Y) {Z : H} (h✝ : CategoryTheory.Limits.colimit ((CategoryTheory.CostructuredArrow.proj L Y).comp F) ⟶ Z) : CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app g.left) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.right E).map g.hom) (CategoryTheory.CategoryStruct.comp h.isoColimit.hom h✝)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.CostructuredArrow.proj L Y).comp F) g) h✝
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.ι_isoColimit_inv_assoc 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) [CategoryTheory.Limits.HasColimit ((CategoryTheory.CostructuredArrow.proj L Y).comp F)] (g : CategoryTheory.CostructuredArrow L Y) {Z : H} (h✝ : (CategoryTheory.StructuredArrow.right E).obj Y ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimit.ι ((CategoryTheory.CostructuredArrow.proj L Y).comp F) g) (CategoryTheory.CategoryStruct.comp h.isoColimit.inv h✝) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app g.left) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.right E).map g.hom) h✝)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.hom_ext' 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) {T : H} {f g : (CategoryTheory.StructuredArrow.right E).obj Y ⟶ T} (hfg : ∀ ⦃X : C⦄ (φ : L.obj X ⟶ Y), CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app X) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.right E).map φ) f) = CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app X) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.right E).map φ) g)) : f = g
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.comp_homEquiv_symm 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) {Z : H} (φ : (CategoryTheory.CostructuredArrow.proj L Y).comp F ⟶ (CategoryTheory.Functor.const (CategoryTheory.CostructuredArrow L Y)).obj Z) (g : CategoryTheory.CostructuredArrow L Y) : CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app g.left) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.right E).map g.hom) ((CategoryTheory.Limits.IsColimit.homEquiv h).symm φ)) = φ.app g
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.comp_homEquiv_symm_assoc 📋 Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
  {C : Type u_1} {D : Type u_2} {H : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_4, u_4} H] {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor C H} {E : L.LeftExtension F} {Y : D} (h : E.IsPointwiseLeftKanExtensionAt Y) {Z : H} (φ : (CategoryTheory.CostructuredArrow.proj L Y).comp F ⟶ (CategoryTheory.Functor.const (CategoryTheory.CostructuredArrow L Y)).obj Z) (g : CategoryTheory.CostructuredArrow L Y) {Z✝ : H} (h✝ : Z ⟶ Z✝) : CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.hom E).app g.left) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.StructuredArrow.right E).map g.hom) (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Limits.IsColimit.homEquiv h).symm φ) h✝)) = CategoryTheory.CategoryStruct.comp (φ.app g) h✝
• CategoryTheory.Presheaf.instUniqueHomLeftExtensionOppositeOpObjFunctorTypeUliftYonedaMkUliftYonedaMap 📋 Mathlib.CategoryTheory.Limits.Presheaf
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (X : C) (Y : F.op.LeftExtension (CategoryTheory.uliftYoneda.{max w v₂, v₁, u₁}.obj X)) : Unique (CategoryTheory.Functor.LeftExtension.mk (CategoryTheory.uliftYoneda.{max w v₁, v₂, u₂}.obj (F.obj X)) (CategoryTheory.uliftYonedaMap F X) ⟶ Y)
• CategoryTheory.Presheaf.instUniqueHomLeftExtensionOppositeOpObjFunctorTypeYonedaMkYonedaMap 📋 Mathlib.CategoryTheory.Limits.Presheaf
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] (F : CategoryTheory.Functor C D) (X : C) (Y : F.op.LeftExtension (CategoryTheory.yoneda.obj X)) : Unique (CategoryTheory.Functor.LeftExtension.mk (CategoryTheory.yoneda.obj (F.obj X)) (CategoryTheory.yonedaMap F X) ⟶ Y)
• CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.extensionHom 📋 Mathlib.CategoryTheory.Limits.Presheaf
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} [∀ (P : CategoryTheory.Functor Cᵒᵖ (Type (max w v₁ v₂))), F.op.HasLeftKanExtension P] (Φ : CategoryTheory.uliftYoneda.{max w v₂, v₁, u₁}.LeftExtension (F.comp CategoryTheory.uliftYoneda.{max w v₁, v₂, u₂})) : CategoryTheory.Functor.LeftExtension.mk F.op.lan (CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan F).hom ⟶ Φ
• CategoryTheory.Presheaf.instUniqueHomLeftExtensionFunctorOppositeTypeUliftYonedaCompMkLanOpHomCompULiftYonedaIsoULiftYonedaCompLan 📋 Mathlib.CategoryTheory.Limits.Presheaf
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) [∀ (P : CategoryTheory.Functor Cᵒᵖ (Type (max w v₁ v₂))), F.op.HasLeftKanExtension P] (Φ : CategoryTheory.StructuredArrow (F.comp CategoryTheory.uliftYoneda.{max w v₁, v₂, u₂}) ((CategoryTheory.Functor.whiskeringLeft C (CategoryTheory.Functor Cᵒᵖ (Type (max w v₁ v₂))) (CategoryTheory.Functor Dᵒᵖ (Type (max w v₁ v₂)))).obj CategoryTheory.uliftYoneda.{max w v₂, v₁, u₁})) : Unique (CategoryTheory.Functor.LeftExtension.mk F.op.lan (CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan F).hom ⟶ Φ)
• CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.hom_ext 📋 Mathlib.CategoryTheory.Limits.Presheaf
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} [∀ (P : CategoryTheory.Functor Cᵒᵖ (Type (max w v₁ v₂))), F.op.HasLeftKanExtension P] {Φ : CategoryTheory.uliftYoneda.{max w v₂, v₁, u₁}.LeftExtension (F.comp CategoryTheory.uliftYoneda.{max w v₁, v₂, u₂})} (f g : CategoryTheory.Functor.LeftExtension.mk F.op.lan (CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan F).hom ⟶ Φ) : f = g
• CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan.hom_ext_iff 📋 Mathlib.CategoryTheory.Limits.Presheaf
  {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} [∀ (P : CategoryTheory.Functor Cᵒᵖ (Type (max w v₁ v₂))), F.op.HasLeftKanExtension P] {Φ : CategoryTheory.uliftYoneda.{max w v₂, v₁, u₁}.LeftExtension (F.comp CategoryTheory.uliftYoneda.{max w v₁, v₂, u₂})} {f g : CategoryTheory.Functor.LeftExtension.mk F.op.lan (CategoryTheory.Presheaf.compULiftYonedaIsoULiftYonedaCompLan F).hom ⟶ Φ} : f = g ↔ True
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.postcompose 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [CategoryTheory.Category.{v_1, u_1} A] [CategoryTheory.Category.{v_2, u_2} B] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) {F : CategoryTheory.Functor A B} {L : CategoryTheory.Functor A C} [G.PreservesPointwiseLeftKanExtension F L] {E : L.LeftExtension F} (hE : E.IsPointwiseLeftKanExtension) : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj E).IsPointwiseLeftKanExtension
• CategoryTheory.Functor.PreservesPointwiseLeftKanExtensionAt.mk' 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [CategoryTheory.Category.{v_1, u_1} A] [CategoryTheory.Category.{v_2, u_2} B] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) (F : CategoryTheory.Functor A B) (L : CategoryTheory.Functor A C) (c : C) {E : L.LeftExtension F} (hE : E.IsPointwiseLeftKanExtensionAt c) (hGE : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj E).IsPointwiseLeftKanExtensionAt c) : G.PreservesPointwiseLeftKanExtensionAt F L c
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.postcompose 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [CategoryTheory.Category.{v_1, u_1} A] [CategoryTheory.Category.{v_2, u_2} B] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) {F : CategoryTheory.Functor A B} {L : CategoryTheory.Functor A C} {c : C} [G.PreservesPointwiseLeftKanExtensionAt F L c] {E : L.LeftExtension F} (hE : E.IsPointwiseLeftKanExtensionAt c) : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj E).IsPointwiseLeftKanExtensionAt c
• CategoryTheory.Functor.PreservesPointwiseLeftKanExtensionAt.mk 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [CategoryTheory.Category.{v_1, u_1} A] [CategoryTheory.Category.{v_2, u_2} B] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Category.{v_4, u_4} D] {G : CategoryTheory.Functor B D} {F : CategoryTheory.Functor A B} {L : CategoryTheory.Functor A C} {c : C} (preserves : ∀ (E : L.LeftExtension F) (a : E.IsPointwiseLeftKanExtensionAt c), Nonempty (((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj E).IsPointwiseLeftKanExtensionAt c)) : G.PreservesPointwiseLeftKanExtensionAt F L c
• CategoryTheory.Functor.PreservesPointwiseLeftKanExtensionAt.preserves 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} {inst✝ : CategoryTheory.Category.{v_1, u_1} A} {inst✝¹ : CategoryTheory.Category.{v_2, u_2} B} {inst✝² : CategoryTheory.Category.{v_3, u_3} C} {inst✝³ : CategoryTheory.Category.{v_4, u_4} D} {G : CategoryTheory.Functor B D} {F : CategoryTheory.Functor A B} {L : CategoryTheory.Functor A C} {c : C} [self : G.PreservesPointwiseLeftKanExtensionAt F L c] (E : L.LeftExtension F) : ∀ (a : E.IsPointwiseLeftKanExtensionAt c), Nonempty (((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj E).IsPointwiseLeftKanExtensionAt c)
• CategoryTheory.Functor.LeftExtension.coconeAtWhiskerRightIso 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [CategoryTheory.Category.{v_1, u_1} A] [CategoryTheory.Category.{v_2, u_2} B] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) (F : CategoryTheory.Functor A B) (L : CategoryTheory.Functor A C) (E : L.LeftExtension F) (c : C) : ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj E).coconeAt c ≅ G.mapCocone (E.coconeAt c)
• CategoryTheory.Functor.PreservesLeftKanExtension.mk' 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [CategoryTheory.Category.{v_1, u_1} A] [CategoryTheory.Category.{v_2, u_2} B] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) (F : CategoryTheory.Functor A B) (L : CategoryTheory.Functor A C) (preserves : ∀ {E : L.LeftExtension F} (a : CategoryTheory.StructuredArrow.IsUniversal E), Nonempty (CategoryTheory.StructuredArrow.IsUniversal ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj E))) : G.PreservesLeftKanExtension F L
• CategoryTheory.Functor.PreservesLeftKanExtension.mk_of_preserves_isUniversal 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [CategoryTheory.Category.{v_1, u_1} A] [CategoryTheory.Category.{v_2, u_2} B] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) (F : CategoryTheory.Functor A B) (L : CategoryTheory.Functor A C) (E : L.LeftExtension F) (hE : CategoryTheory.StructuredArrow.IsUniversal E) (h : Nonempty (CategoryTheory.StructuredArrow.IsUniversal ((CategoryTheory.Functor.LeftExtension.postcompose₂ L F G).obj E))) : G.PreservesLeftKanExtension F L
• CategoryTheory.Functor.LeftExtension.coconeAtWhiskerRightIso_hom_hom 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [CategoryTheory.Category.{v_1, u_1} A] [CategoryTheory.Category.{v_2, u_2} B] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) (F : CategoryTheory.Functor A B) (L : CategoryTheory.Functor A C) (E : L.LeftExtension F) (c : C) : (CategoryTheory.Functor.LeftExtension.coconeAtWhiskerRightIso G F L E c).hom.hom = CategoryTheory.CategoryStruct.id (G.obj (E.right.obj c))
• CategoryTheory.Functor.LeftExtension.coconeAtWhiskerRightIso_inv_hom 📋 Mathlib.CategoryTheory.Functor.KanExtension.Preserves
  {A : Type u_1} {B : Type u_2} {C : Type u_3} {D : Type u_4} [CategoryTheory.Category.{v_1, u_1} A] [CategoryTheory.Category.{v_2, u_2} B] [CategoryTheory.Category.{v_3, u_3} C] [CategoryTheory.Category.{v_4, u_4} D] (G : CategoryTheory.Functor B D) (F : CategoryTheory.Functor A B) (L : CategoryTheory.Functor A C) (E : L.LeftExtension F) (c : C) : (CategoryTheory.Functor.LeftExtension.coconeAtWhiskerRightIso G F L E c).inv.hom = CategoryTheory.CategoryStruct.id (G.obj (E.right.obj c))
• CategoryTheory.Adjunction.leftExtensionPostCompose₂RightAdjoint 📋 Mathlib.CategoryTheory.Functor.KanExtension.AdjunctionPreserves
  {C : Type u_1} {D : Type u_2} {H₁ : Type u_3} {H₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H₁] [CategoryTheory.Category.{v_4, u_4} H₂] {G₁ : CategoryTheory.Functor H₁ H₂} {G₂ : CategoryTheory.Functor H₂ H₁} (adj : G₁ ⊣ G₂) (F : CategoryTheory.Functor C H₁) (L : CategoryTheory.Functor C D) : CategoryTheory.Functor (L.LeftExtension (F.comp G₁)) (L.LeftExtension F)
• CategoryTheory.Adjunction.leftExtensionPostcompose₂ 📋 Mathlib.CategoryTheory.Functor.KanExtension.AdjunctionPreserves
  {C : Type u_1} {D : Type u_2} {H₁ : Type u_3} {H₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H₁] [CategoryTheory.Category.{v_4, u_4} H₂] {G₁ : CategoryTheory.Functor H₁ H₂} {G₂ : CategoryTheory.Functor H₂ H₁} (adj : G₁ ⊣ G₂) (F : CategoryTheory.Functor C H₁) (L : CategoryTheory.Functor C D) : CategoryTheory.Functor.LeftExtension.postcompose₂ L F G₁ ⊣ adj.leftExtensionPostCompose₂RightAdjoint F L
• CategoryTheory.Adjunction.leftExtensionPostCompose₂RightAdjoint_obj_right_obj 📋 Mathlib.CategoryTheory.Functor.KanExtension.AdjunctionPreserves
  {C : Type u_1} {D : Type u_2} {H₁ : Type u_3} {H₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H₁] [CategoryTheory.Category.{v_4, u_4} H₂] {G₁ : CategoryTheory.Functor H₁ H₂} {G₂ : CategoryTheory.Functor H₂ H₁} (adj : G₁ ⊣ G₂) (F : CategoryTheory.Functor C H₁) (L : CategoryTheory.Functor C D) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit (F.comp G₁)) ((CategoryTheory.Functor.whiskeringLeft C D H₂).obj L)) (X✝ : D) : ((adj.leftExtensionPostCompose₂RightAdjoint F L).obj X).right.obj X✝ = G₂.obj (X.right.obj X✝)
• CategoryTheory.Adjunction.leftExtensionPostCompose₂RightAdjoint_obj_right_map 📋 Mathlib.CategoryTheory.Functor.KanExtension.AdjunctionPreserves
  {C : Type u_1} {D : Type u_2} {H₁ : Type u_3} {H₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H₁] [CategoryTheory.Category.{v_4, u_4} H₂] {G₁ : CategoryTheory.Functor H₁ H₂} {G₂ : CategoryTheory.Functor H₂ H₁} (adj : G₁ ⊣ G₂) (F : CategoryTheory.Functor C H₁) (L : CategoryTheory.Functor C D) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit (F.comp G₁)) ((CategoryTheory.Functor.whiskeringLeft C D H₂).obj L)) {X✝ Y✝ : D} (f : X✝ ⟶ Y✝) : ((adj.leftExtensionPostCompose₂RightAdjoint F L).obj X).right.map f = G₂.map (X.right.map f)
• CategoryTheory.Adjunction.leftExtensionPostCompose₂RightAdjoint_obj_hom_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.AdjunctionPreserves
  {C : Type u_1} {D : Type u_2} {H₁ : Type u_3} {H₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H₁] [CategoryTheory.Category.{v_4, u_4} H₂] {G₁ : CategoryTheory.Functor H₁ H₂} {G₂ : CategoryTheory.Functor H₂ H₁} (adj : G₁ ⊣ G₂) (F : CategoryTheory.Functor C H₁) (L : CategoryTheory.Functor C D) (X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit (F.comp G₁)) ((CategoryTheory.Functor.whiskeringLeft C D H₂).obj L)) (X✝ : C) : ((adj.leftExtensionPostCompose₂RightAdjoint F L).obj X).hom.app X✝ = CategoryTheory.CategoryStruct.comp (adj.unit.app (F.obj X✝)) (G₂.map (X.hom.app X✝))
• CategoryTheory.Adjunction.leftExtensionPostCompose₂RightAdjoint_map_right_app 📋 Mathlib.CategoryTheory.Functor.KanExtension.AdjunctionPreserves
  {C : Type u_1} {D : Type u_2} {H₁ : Type u_3} {H₂ : Type u_4} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} D] [CategoryTheory.Category.{v_3, u_3} H₁] [CategoryTheory.Category.{v_4, u_4} H₂] {G₁ : CategoryTheory.Functor H₁ H₂} {G₂ : CategoryTheory.Functor H₂ H₁} (adj : G₁ ⊣ G₂) (F : CategoryTheory.Functor C H₁) (L : CategoryTheory.Functor C D) {X Y : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit (F.comp G₁)) ((CategoryTheory.Functor.whiskeringLeft C D H₂).obj L)} (φ : X ⟶ Y) (X✝ : D) : ((adj.leftExtensionPostCompose₂RightAdjoint F L).map φ).right.app X✝ = G₂.map (φ.right.app X✝)
• CategoryTheory.Functor.LeftExtension.isPointwiseLeftKanExtensionOfIsIsoOfIsLocalization 📋 Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived
  {C : Type u₁} {D : Type u₂} {H : Type u₃} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Category.{v₃, u₃} H] {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (W : CategoryTheory.MorphismProperty C) (E : L.LeftExtension F) [CategoryTheory.IsIso (CategoryTheory.StructuredArrow.hom E)] [L.IsLocalization W] : E.IsPointwiseLeftKanExtension
• CategoryTheory.Functor.LeftExtension.compTwoSquare 📋 Mathlib.CategoryTheory.GuitartExact.KanExtension
  {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {C₄ : Type u₄} {D : Type u₅} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} C₃] [CategoryTheory.Category.{v₄, u₄} C₄] [CategoryTheory.Category.{v₅, u₅} D] {T : CategoryTheory.Functor C₁ C₂} {L : CategoryTheory.Functor C₁ C₃} {R : CategoryTheory.Functor C₂ C₄} {B : CategoryTheory.Functor C₃ C₄} {F : CategoryTheory.Functor C₂ D} (E : R.LeftExtension F) (w : CategoryTheory.TwoSquare T L R B) : L.LeftExtension (T.comp F)
• CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.compTwoSquare 📋 Mathlib.CategoryTheory.GuitartExact.KanExtension
  {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {C₄ : Type u₄} {D : Type u₅} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} C₃] [CategoryTheory.Category.{v₄, u₄} C₄] [CategoryTheory.Category.{v₅, u₅} D] {T : CategoryTheory.Functor C₁ C₂} {L : CategoryTheory.Functor C₁ C₃} {R : CategoryTheory.Functor C₂ C₄} {B : CategoryTheory.Functor C₃ C₄} {F : CategoryTheory.Functor C₂ D} {E : R.LeftExtension F} (h : E.IsPointwiseLeftKanExtension) (w : CategoryTheory.TwoSquare T L R B) [w.GuitartExact] : (E.compTwoSquare w).IsPointwiseLeftKanExtension
• CategoryTheory.Functor.LeftExtension.isPointwiseLeftKanExtensionOfCompTwoSquare 📋 Mathlib.CategoryTheory.GuitartExact.KanExtension
  {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {C₄ : Type u₄} {D : Type u₅} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} C₃] [CategoryTheory.Category.{v₄, u₄} C₄] [CategoryTheory.Category.{v₅, u₅} D] {T : CategoryTheory.Functor C₁ C₂} {L : CategoryTheory.Functor C₁ C₃} {R : CategoryTheory.Functor C₂ C₄} {B : CategoryTheory.Functor C₃ C₄} {F : CategoryTheory.Functor C₂ D} (E : R.LeftExtension F) (w : CategoryTheory.TwoSquare T L R B) [w.GuitartExact] [B.EssSurj] (h : (E.compTwoSquare w).IsPointwiseLeftKanExtension) : E.IsPointwiseLeftKanExtension
• CategoryTheory.Functor.LeftExtension.isPointwiseLeftKanExtensionEquivOfGuitartExact 📋 Mathlib.CategoryTheory.GuitartExact.KanExtension
  {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {C₄ : Type u₄} {D : Type u₅} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} C₃] [CategoryTheory.Category.{v₄, u₄} C₄] [CategoryTheory.Category.{v₅, u₅} D] {T : CategoryTheory.Functor C₁ C₂} {L : CategoryTheory.Functor C₁ C₃} {R : CategoryTheory.Functor C₂ C₄} {B : CategoryTheory.Functor C₃ C₄} {F : CategoryTheory.Functor C₂ D} (E : R.LeftExtension F) (w : CategoryTheory.TwoSquare T L R B) [w.GuitartExact] [B.EssSurj] : (E.compTwoSquare w).IsPointwiseLeftKanExtension ≃ E.IsPointwiseLeftKanExtension
• CategoryTheory.Functor.LeftExtension.isPointwiseLeftKanExtensionAtCompTwoSquareEquiv 📋 Mathlib.CategoryTheory.GuitartExact.KanExtension
  {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {C₄ : Type u₄} {D : Type u₅} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} C₃] [CategoryTheory.Category.{v₄, u₄} C₄] [CategoryTheory.Category.{v₅, u₅} D] {T : CategoryTheory.Functor C₁ C₂} {L : CategoryTheory.Functor C₁ C₃} {R : CategoryTheory.Functor C₂ C₄} {B : CategoryTheory.Functor C₃ C₄} {F : CategoryTheory.Functor C₂ D} (E : R.LeftExtension F) (w : CategoryTheory.TwoSquare T L R B) (X₃ : C₃) [(w.costructuredArrowRightwards X₃).Final] : (E.compTwoSquare w).IsPointwiseLeftKanExtensionAt X₃ ≃ E.IsPointwiseLeftKanExtensionAt (B.obj X₃)
• CategoryTheory.Functor.LeftExtension.nonempty_isPointwiseLeftKanExtensionAt_compTwoSquare_iff 📋 Mathlib.CategoryTheory.GuitartExact.KanExtension
  {C₁ : Type u₁} {C₂ : Type u₂} {C₃ : Type u₃} {C₄ : Type u₄} {D : Type u₅} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] [CategoryTheory.Category.{v₃, u₃} C₃] [CategoryTheory.Category.{v₄, u₄} C₄] [CategoryTheory.Category.{v₅, u₅} D] {T : CategoryTheory.Functor C₁ C₂} {L : CategoryTheory.Functor C₁ C₃} {R : CategoryTheory.Functor C₂ C₄} {B : CategoryTheory.Functor C₃ C₄} {F : CategoryTheory.Functor C₂ D} (E : R.LeftExtension F) (w : CategoryTheory.TwoSquare T L R B) (X₃ : C₃) [(w.costructuredArrowRightwards X₃).Final] : Nonempty ((E.compTwoSquare w).IsPointwiseLeftKanExtensionAt X₃) ↔ Nonempty (E.IsPointwiseLeftKanExtensionAt (B.obj X₃))

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