File: Mathlib/CategoryTheory/Idempotents/HomologicalComplex.lean

full_name: CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comp_d
code: @[simp, reassoc]
theorem p_comp_d : P.p.f n ≫ f.f.f n = f.f.f n 
goal: CategoryTheory.CategoryStruct.comp (P.p.f n) (f.f.f n) = f.f.f n

full_name: CategoryTheory.Idempotents.Karoubi.HomologicalComplex.comp_p_d
code: @[simp, reassoc]
theorem comp_p_d : f.f.f n ≫ Q.p.f n = f.f.f n 
goal: CategoryTheory.CategoryStruct.comp (f.f.f n) (Q.p.f n) = f.f.f n

full_name: CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comm_f
code: @[reassoc]
theorem p_comm_f : P.p.f n ≫ f.f.f n = f.f.f n ≫ Q.p.f n 
goal: CategoryTheory.CategoryStruct.comp (P.p.f n) (f.f.f n) = CategoryTheory.CategoryStruct.comp (f.f.f n) (Q.p.f n)

full_name: CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_idem
code: @[simp, reassoc]
theorem p_idem : P.p.f n ≫ P.p.f n = P.p.f n 
goal: CategoryTheory.CategoryStruct.comp (P.p.f n) (P.p.f n) = P.p.f n

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj_d_f
code: /-- The functor `Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c`,
on objects. -/
@[simps]
def obj (P : Karoubi (HomologicalComplex C c)) : HomologicalComplex (Karoubi C) c 
goal: ((CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj P).d i j).f =
  CategoryTheory.CategoryStruct.comp (P.p.f i) (P.X.d i j)

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj
code: /-- The functor `Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c`,
on objects. -/
@[simps]
def obj (P : Karoubi (HomologicalComplex C c)) : HomologicalComplex (Karoubi C) c 
goal: HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj_X_p
code: /-- The functor `Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c`,
on objects. -/
@[simps]
def obj (P : Karoubi (HomologicalComplex C c)) : HomologicalComplex (Karoubi C) c 
goal: ((CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj P).X n).p = P.p.f n

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj_X_X
code: /-- The functor `Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c`,
on objects. -/
@[simps]
def obj (P : Karoubi (HomologicalComplex C c)) : HomologicalComplex (Karoubi C) c 
goal: ((CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj P).X n).X = P.X.X n

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.map_f_f
code: /-- The functor `Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c`,
on morphisms. -/
@[simps]
def map {P Q : Karoubi (HomologicalComplex C c)} (f : P ⟶ Q) : obj P ⟶ obj Q 
goal: ((CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.map f).f n).f = f.f.f n

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.map
code: /-- The functor `Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c`,
on morphisms. -/
@[simps]
def map {P Q : Karoubi (HomologicalComplex C c)} (f : P ⟶ Q) : obj P ⟶ obj Q 
goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj P ⟶
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj Q

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor_map
code: /-- The functor `Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def functor : Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c 
goal: ∀ {X Y : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)} (f : X ⟶ Y),
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor.map f =
    CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.map f

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor
code: /-- The functor `Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def functor : Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c 
goal: CategoryTheory.Functor (CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c))
  (HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c)

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor_obj
code: /-- The functor `Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def functor : Karoubi (HomologicalComplex C c) ⥤ HomologicalComplex (Karoubi C) c 
goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor.obj P =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Functor.obj P

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj
code: /-- The functor `HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c)`,
on objects -/
@[simps]
def obj (K : HomologicalComplex (Karoubi C) c) : Karoubi (HomologicalComplex C c) 
goal: CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj_p_f
code: /-- The functor `HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c)`,
on objects -/
@[simps]
def obj (K : HomologicalComplex (Karoubi C) c) : Karoubi (HomologicalComplex C c) 
goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj K).p.f n = (K.X n).p

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj_X_X
code: /-- The functor `HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c)`,
on objects -/
@[simps]
def obj (K : HomologicalComplex (Karoubi C) c) : Karoubi (HomologicalComplex C c) 
goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj K).X.X n = (K.X n).X

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj_X_d
code: /-- The functor `HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c)`,
on objects -/
@[simps]
def obj (K : HomologicalComplex (Karoubi C) c) : Karoubi (HomologicalComplex C c) 
goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj K).X.d i j = (K.d i j).f

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.map
code: /-- The functor `HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c)`,
on morphisms -/
@[simps]
def map {K L : HomologicalComplex (Karoubi C) c} (f : K ⟶ L) : obj K ⟶ obj L 
goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj K ⟶
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj L

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.map_f_f
code: /-- The functor `HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c)`,
on morphisms -/
@[simps]
def map {K L : HomologicalComplex (Karoubi C) c} (f : K ⟶ L) : obj K ⟶ obj L 
goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.map f).f.f n = (f.f n).f

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse_map
code: /-- The functor `HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c)`. -/
@[simps]
def inverse : HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c) 
goal: ∀ {X Y : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c} (f : X ⟶ Y),
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse.map f =
    CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.map f

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse
code: /-- The functor `HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c)`. -/
@[simps]
def inverse : HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c) 
goal: CategoryTheory.Functor (HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c)
  (CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c))

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse_obj
code: /-- The functor `HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c)`. -/
@[simps]
def inverse : HomologicalComplex (Karoubi C) c ⥤ Karoubi (HomologicalComplex C c) 
goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse.obj K =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.Inverse.obj K

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso
code: /-- The counit isomorphism of the equivalence
`Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (HomologicalComplex (Karoubi C) c) 
goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse.comp
    CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor ≅
  CategoryTheory.Functor.id (HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c)

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso_hom
code: /-- The counit isomorphism of the equivalence
`Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (HomologicalComplex (Karoubi C) c) 
goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso.hom = CategoryTheory.eqToHom ⋯

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso_inv
code: /-- The counit isomorphism of the equivalence
`Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (HomologicalComplex (Karoubi C) c) 
goal: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso.inv = CategoryTheory.eqToHom ⋯

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso
code: /-- The unit isomorphism of the equivalence
`Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def unitIso : 𝟭 (Karoubi (HomologicalComplex C c)) ≅ functor ⋙ inverse 
goal: CategoryTheory.Functor.id (CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c)) ≅
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor.comp
    CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso_hom_app_f_f
code: /-- The unit isomorphism of the equivalence
`Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def unitIso : 𝟭 (Karoubi (HomologicalComplex C c)) ≅ functor ⋙ inverse 
goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso.hom.app P).f.f n = P.p.f n

full_name: CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso_inv_app_f_f
code: /-- The unit isomorphism of the equivalence
`Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def unitIso : 𝟭 (Karoubi (HomologicalComplex C c)) ≅ functor ⋙ inverse 
goal: (CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso.inv.app P).f.f n = P.p.f n

full_name: CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence
code: /-- The equivalence `Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def karoubiHomologicalComplexEquivalence :
    Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c 
goal: CategoryTheory.Idempotents.Karoubi (HomologicalComplex C c) ≌
  HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) c

full_name: CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_counitIso
code: /-- The equivalence `Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def karoubiHomologicalComplexEquivalence :
    Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c 
goal: (CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence C c).counitIso =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.counitIso

full_name: CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_unitIso
code: /-- The equivalence `Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def karoubiHomologicalComplexEquivalence :
    Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c 
goal: (CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence C c).unitIso =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.unitIso

full_name: CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_functor
code: /-- The equivalence `Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def karoubiHomologicalComplexEquivalence :
    Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c 
goal: (CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence C c).functor =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.functor

full_name: CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence_inverse
code: /-- The equivalence `Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c`. -/
@[simps]
def karoubiHomologicalComplexEquivalence :
    Karoubi (HomologicalComplex C c) ≌ HomologicalComplex (Karoubi C) c 
goal: (CategoryTheory.Idempotents.karoubiHomologicalComplexEquivalence C c).inverse =
  CategoryTheory.Idempotents.KaroubiHomologicalComplexEquivalence.inverse

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_obj_X_X
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: (((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).functor.obj P).X n).X = P.X.X n

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_map_f_f
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: ∀ {X Y : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) (ComplexShape.down α)} (f : X ⟶ Y) (n : α),
  ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).inverse.map f).f.f n = (f.f n).f

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_map_f_f
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: ∀ {X Y : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C (ComplexShape.down α))} (f : X ⟶ Y) (n : α),
  (((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).functor.map f).f n).f = f.f.f n

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_obj_p_f
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).inverse.obj K).p.f n = (K.X n).p

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_obj_X_d
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).inverse.obj K).X.d i j = (K.d i j).f

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_hom_app_f_f
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).unitIso.hom.app P).f.f n = P.p.f n

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_obj_X_p
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: (((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).functor.obj P).X n).p = P.p.f n

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_unitIso_inv_app_f_f
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).unitIso.inv.app P).f.f n = P.p.f n

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_functor_obj_d_f
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: (((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).functor.obj P).d i j).f =
  CategoryTheory.CategoryStruct.comp (P.X.d i j) (P.p.f j)

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_hom
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: (CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).counitIso.hom = CategoryTheory.eqToHom ⋯

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_inverse_obj_X_X
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).inverse.obj K).X.X n = (K.X n).X

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: CategoryTheory.Idempotents.Karoubi (ChainComplex C α) ≌ ChainComplex (CategoryTheory.Idempotents.Karoubi C) α

full_name: CategoryTheory.Idempotents.karoubiChainComplexEquivalence_counitIso_inv
code: /-- The equivalence `Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiChainComplexEquivalence : Karoubi (ChainComplex C α) ≌ ChainComplex (Karoubi C) α 
goal: (CategoryTheory.Idempotents.karoubiChainComplexEquivalence C α).counitIso.inv = CategoryTheory.eqToHom ⋯

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_map_f_f
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: ∀ {X Y : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) (ComplexShape.up α)} (f : X ⟶ Y) (n : α),
  ((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).inverse.map f).f.f n = (f.f n).f

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_counitIso_inv
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: (CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).counitIso.inv = CategoryTheory.eqToHom ⋯

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_obj_p_f
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).inverse.obj K).p.f n = (K.X n).p

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_functor_obj_X_X
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: (((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).functor.obj P).X n).X = P.X.X n

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_unitIso_hom_app_f_f
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).unitIso.hom.app P).f.f n = P.p.f n

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_counitIso_hom
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: (CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).counitIso.hom = CategoryTheory.eqToHom ⋯

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_unitIso_inv_app_f_f
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).unitIso.inv.app P).f.f n = P.p.f n

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_functor_obj_X_p
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: (((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).functor.obj P).X n).p = P.p.f n

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_obj_X_X
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).inverse.obj K).X.X n = (K.X n).X

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_inverse_obj_X_d
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: ((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).inverse.obj K).X.d i j = (K.d i j).f

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: CategoryTheory.Idempotents.Karoubi (CochainComplex C α) ≌ CochainComplex (CategoryTheory.Idempotents.Karoubi C) α

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_functor_obj_d_f
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: (((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).functor.obj P).d i j).f =
  CategoryTheory.CategoryStruct.comp (P.X.d i j) (P.p.f j)

full_name: CategoryTheory.Idempotents.karoubiCochainComplexEquivalence_functor_map_f_f
code: /-- The equivalence `Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α`. -/
@[simps!]
def karoubiCochainComplexEquivalence :
    Karoubi (CochainComplex C α) ≌ CochainComplex (Karoubi C) α 
goal: ∀ {X Y : CategoryTheory.Idempotents.Karoubi (HomologicalComplex C (ComplexShape.up α))} (f : X ⟶ Y) (n : α),
  (((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).functor.map f).f n).f = f.f.f n

full_name: CategoryTheory.Idempotents.instIsIdempotentCompleteHomologicalComplex
code: instance [IsIdempotentComplete C] : IsIdempotentComplete (HomologicalComplex C c) 
goal: CategoryTheory.IsIdempotentComplete (HomologicalComplex C c)