File: Mathlib/Algebra/Homology/ShortComplex/FunctorEquivalence.lean

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.functor_obj_map
code: /-- The obvious functor `ShortComplex (J ⥤ C) ⥤ J ⥤ ShortComplex C`. -/
@[simps]
def functor : ShortComplex (J ⥤ C) ⥤ J ⥤ ShortComplex C 
goal: ∀ {X Y : J} (f : X ⟶ Y),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.functor J C).obj S).map f =
    S.mapNatTrans ((CategoryTheory.evaluation J C).map f)

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.functor
code: /-- The obvious functor `ShortComplex (J ⥤ C) ⥤ J ⥤ ShortComplex C`. -/
@[simps]
def functor : ShortComplex (J ⥤ C) ⥤ J ⥤ ShortComplex C 
goal: CategoryTheory.Functor (CategoryTheory.ShortComplex (CategoryTheory.Functor J C))
  (CategoryTheory.Functor J (CategoryTheory.ShortComplex C))

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.functor_obj_obj
code: /-- The obvious functor `ShortComplex (J ⥤ C) ⥤ J ⥤ ShortComplex C`. -/
@[simps]
def functor : ShortComplex (J ⥤ C) ⥤ J ⥤ ShortComplex C 
goal: ((CategoryTheory.ShortComplex.FunctorEquivalence.functor J C).obj S).obj j =
  S.map ((CategoryTheory.evaluation J C).obj j)

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.functor_map_app
code: /-- The obvious functor `ShortComplex (J ⥤ C) ⥤ J ⥤ ShortComplex C`. -/
@[simps]
def functor : ShortComplex (J ⥤ C) ⥤ J ⥤ ShortComplex C 
goal: ∀ {X Y : CategoryTheory.ShortComplex (CategoryTheory.Functor J C)} (φ : X ⟶ Y) (j : J),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.functor J C).map φ).app j =
    ((CategoryTheory.evaluation J C).obj j).mapShortComplex.map φ

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.inverse_obj_g
code: /-- The obvious functor `(J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C)`. -/
@[simps]
def inverse : (J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C) 
goal: ((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).obj F).g =
  CategoryTheory.whiskerLeft F CategoryTheory.ShortComplex.π₂Toπ₃

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.inverse_obj_X₃
code: /-- The obvious functor `(J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C)`. -/
@[simps]
def inverse : (J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C) 
goal: ((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).obj F).X₃ = F.comp CategoryTheory.ShortComplex.π₃

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.inverse_obj_X₂
code: /-- The obvious functor `(J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C)`. -/
@[simps]
def inverse : (J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C) 
goal: ((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).obj F).X₂ = F.comp CategoryTheory.ShortComplex.π₂

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.inverse_obj_f
code: /-- The obvious functor `(J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C)`. -/
@[simps]
def inverse : (J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C) 
goal: ((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).obj F).f =
  CategoryTheory.whiskerLeft F CategoryTheory.ShortComplex.π₁Toπ₂

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.inverse
code: /-- The obvious functor `(J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C)`. -/
@[simps]
def inverse : (J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C) 
goal: CategoryTheory.Functor (CategoryTheory.Functor J (CategoryTheory.ShortComplex C))
  (CategoryTheory.ShortComplex (CategoryTheory.Functor J C))

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.inverse_obj_X₁
code: /-- The obvious functor `(J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C)`. -/
@[simps]
def inverse : (J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C) 
goal: ((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).obj F).X₁ = F.comp CategoryTheory.ShortComplex.π₁

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.inverse_map_τ₂
code: /-- The obvious functor `(J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C)`. -/
@[simps]
def inverse : (J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C) 
goal: ∀ {X Y : CategoryTheory.Functor J (CategoryTheory.ShortComplex C)} (φ : X ⟶ Y),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).map φ).τ₂ =
    CategoryTheory.whiskerRight φ CategoryTheory.ShortComplex.π₂

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.inverse_map_τ₃
code: /-- The obvious functor `(J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C)`. -/
@[simps]
def inverse : (J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C) 
goal: ∀ {X Y : CategoryTheory.Functor J (CategoryTheory.ShortComplex C)} (φ : X ⟶ Y),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).map φ).τ₃ =
    CategoryTheory.whiskerRight φ CategoryTheory.ShortComplex.π₃

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.inverse_map_τ₁
code: /-- The obvious functor `(J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C)`. -/
@[simps]
def inverse : (J ⥤ ShortComplex C) ⥤ ShortComplex (J ⥤ C) 
goal: ∀ {X Y : CategoryTheory.Functor J (CategoryTheory.ShortComplex C)} (φ : X ⟶ Y),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).map φ).τ₁ =
    CategoryTheory.whiskerRight φ CategoryTheory.ShortComplex.π₁

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.unitIso_inv_app_τ₂_app
code: /-- The unit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def unitIso : 𝟭 _ ≅ functor J C ⋙ inverse J C 
goal: ∀ (X_1 : J),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.unitIso J C).inv.app X).τ₂.app X_1 =
    CategoryTheory.CategoryStruct.id (X.X₂.obj X_1)

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.unitIso_inv_app_τ₃_app
code: /-- The unit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def unitIso : 𝟭 _ ≅ functor J C ⋙ inverse J C 
goal: ∀ (X_1 : J),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.unitIso J C).inv.app X).τ₃.app X_1 =
    CategoryTheory.CategoryStruct.id (X.X₃.obj X_1)

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.unitIso_hom_app_τ₁_app
code: /-- The unit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def unitIso : 𝟭 _ ≅ functor J C ⋙ inverse J C 
goal: ∀ (X_1 : J),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.unitIso J C).hom.app X).τ₁.app X_1 =
    CategoryTheory.CategoryStruct.id (X.X₁.obj X_1)

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.unitIso_hom_app_τ₃_app
code: /-- The unit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def unitIso : 𝟭 _ ≅ functor J C ⋙ inverse J C 
goal: ∀ (X_1 : J),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.unitIso J C).hom.app X).τ₃.app X_1 =
    CategoryTheory.CategoryStruct.id (X.X₃.obj X_1)

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.unitIso
code: /-- The unit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def unitIso : 𝟭 _ ≅ functor J C ⋙ inverse J C 
goal: CategoryTheory.Functor.id (CategoryTheory.ShortComplex (CategoryTheory.Functor J C)) ≅
  (CategoryTheory.ShortComplex.FunctorEquivalence.functor J C).comp
    (CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C)

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.unitIso_hom_app_τ₂_app
code: /-- The unit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def unitIso : 𝟭 _ ≅ functor J C ⋙ inverse J C 
goal: ∀ (X_1 : J),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.unitIso J C).hom.app X).τ₂.app X_1 =
    CategoryTheory.CategoryStruct.id (X.X₂.obj X_1)

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.unitIso_inv_app_τ₁_app
code: /-- The unit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def unitIso : 𝟭 _ ≅ functor J C ⋙ inverse J C 
goal: ∀ (X_1 : J),
  ((CategoryTheory.ShortComplex.FunctorEquivalence.unitIso J C).inv.app X).τ₁.app X_1 =
    CategoryTheory.CategoryStruct.id (X.X₁.obj X_1)

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.counitIso_inv_app_app_τ₂
code: /-- The counit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def counitIso : inverse J C ⋙ functor J C ≅ 𝟭 _ 
goal: ∀ (X_1 : J),
  (((CategoryTheory.ShortComplex.FunctorEquivalence.counitIso J C).inv.app X).app X_1).τ₂ =
    CategoryTheory.CategoryStruct.id (X.obj X_1).X₂

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.counitIso_inv_app_app_τ₃
code: /-- The counit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def counitIso : inverse J C ⋙ functor J C ≅ 𝟭 _ 
goal: ∀ (X_1 : J),
  (((CategoryTheory.ShortComplex.FunctorEquivalence.counitIso J C).inv.app X).app X_1).τ₃ =
    CategoryTheory.CategoryStruct.id (X.obj X_1).X₃

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.counitIso_hom_app_app_τ₂
code: /-- The counit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def counitIso : inverse J C ⋙ functor J C ≅ 𝟭 _ 
goal: ∀ (X_1 : J),
  (((CategoryTheory.ShortComplex.FunctorEquivalence.counitIso J C).hom.app X).app X_1).τ₂ =
    CategoryTheory.CategoryStruct.id (X.obj X_1).X₂

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.counitIso
code: /-- The counit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def counitIso : inverse J C ⋙ functor J C ≅ 𝟭 _ 
goal: (CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).comp
    (CategoryTheory.ShortComplex.FunctorEquivalence.functor J C) ≅
  CategoryTheory.Functor.id (CategoryTheory.Functor J (CategoryTheory.ShortComplex C))

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.counitIso_hom_app_app_τ₁
code: /-- The counit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def counitIso : inverse J C ⋙ functor J C ≅ 𝟭 _ 
goal: ∀ (X_1 : J),
  (((CategoryTheory.ShortComplex.FunctorEquivalence.counitIso J C).hom.app X).app X_1).τ₁ =
    CategoryTheory.CategoryStruct.id (X.obj X_1).X₁

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.counitIso_inv_app_app_τ₁
code: /-- The counit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def counitIso : inverse J C ⋙ functor J C ≅ 𝟭 _ 
goal: ∀ (X_1 : J),
  (((CategoryTheory.ShortComplex.FunctorEquivalence.counitIso J C).inv.app X).app X_1).τ₁ =
    CategoryTheory.CategoryStruct.id (X.obj X_1).X₁

full_name: CategoryTheory.ShortComplex.FunctorEquivalence.counitIso_hom_app_app_τ₃
code: /-- The counit isomorphism of the equivalence
`ShortComplex.functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps!]
def counitIso : inverse J C ⋙ functor J C ≅ 𝟭 _ 
goal: ∀ (X_1 : J),
  (((CategoryTheory.ShortComplex.FunctorEquivalence.counitIso J C).hom.app X).app X_1).τ₃ =
    CategoryTheory.CategoryStruct.id (X.obj X_1).X₃

full_name: CategoryTheory.ShortComplex.functorEquivalence
code: /-- The obvious equivalence `ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps]
def functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C 
goal: CategoryTheory.ShortComplex (CategoryTheory.Functor J C) ≌ CategoryTheory.Functor J (CategoryTheory.ShortComplex C)

full_name: CategoryTheory.ShortComplex.functorEquivalence_unitIso
code: /-- The obvious equivalence `ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps]
def functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C 
goal: (CategoryTheory.ShortComplex.functorEquivalence J C).unitIso =
  CategoryTheory.ShortComplex.FunctorEquivalence.unitIso J C

full_name: CategoryTheory.ShortComplex.functorEquivalence_functor
code: /-- The obvious equivalence `ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps]
def functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C 
goal: (CategoryTheory.ShortComplex.functorEquivalence J C).functor =
  CategoryTheory.ShortComplex.FunctorEquivalence.functor J C

full_name: CategoryTheory.ShortComplex.functorEquivalence_counitIso
code: /-- The obvious equivalence `ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps]
def functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C 
goal: (CategoryTheory.ShortComplex.functorEquivalence J C).counitIso =
  CategoryTheory.ShortComplex.FunctorEquivalence.counitIso J C

full_name: CategoryTheory.ShortComplex.functorEquivalence_inverse
code: /-- The obvious equivalence `ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C`. -/
@[simps]
def functorEquivalence : ShortComplex (J ⥤ C) ≌ J ⥤ ShortComplex C 
goal: (CategoryTheory.ShortComplex.functorEquivalence J C).inverse =
  CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C