File: Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.lean

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.idem_f
code: @[reassoc (attr := simp)]
lemma idem_f (P : Karoubi (Karoubi C)) : P.p.f ≫ P.p.f = P.p.f 
goal: CategoryTheory.CategoryStruct.comp P.p.f P.p.f = P.p.f

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.p_comm_f
code: @[reassoc]
lemma p_comm_f {P Q : Karoubi (Karoubi C)} (f : P ⟶ Q) : P.p.f ≫ f.f.f = f.f.f ≫ Q.p.f 
goal: CategoryTheory.CategoryStruct.comp P.p.f f.f.f = CategoryTheory.CategoryStruct.comp f.f.f Q.p.f

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.inverse_obj_p
code: /-- The canonical functor `Karoubi (Karoubi C) ⥤ Karoubi C` -/
@[simps]
def inverse : Karoubi (Karoubi C) ⥤ Karoubi C 
goal: ((CategoryTheory.Idempotents.KaroubiKaroubi.inverse C).obj P).p = P.p.f

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.inverse_obj_X
code: /-- The canonical functor `Karoubi (Karoubi C) ⥤ Karoubi C` -/
@[simps]
def inverse : Karoubi (Karoubi C) ⥤ Karoubi C 
goal: ((CategoryTheory.Idempotents.KaroubiKaroubi.inverse C).obj P).X = P.X.X

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

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

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.instAdditiveKaroubiInverse
code: instance [Preadditive C] : Functor.Additive (inverse C) 
goal: (CategoryTheory.Idempotents.KaroubiKaroubi.inverse C).Additive

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.unitIso_inv_app_f
code: /-- The unit isomorphism of the equivalence -/
@[simps!]
def unitIso : 𝟭 (Karoubi C) ≅ toKaroubi (Karoubi C) ⋙ inverse C 
goal: ((CategoryTheory.Idempotents.KaroubiKaroubi.unitIso C).inv.app X).f = X.p

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.unitIso
code: /-- The unit isomorphism of the equivalence -/
@[simps!]
def unitIso : 𝟭 (Karoubi C) ≅ toKaroubi (Karoubi C) ⋙ inverse C 
goal: CategoryTheory.Functor.id (CategoryTheory.Idempotents.Karoubi C) ≅
  (CategoryTheory.Idempotents.toKaroubi (CategoryTheory.Idempotents.Karoubi C)).comp
    (CategoryTheory.Idempotents.KaroubiKaroubi.inverse C)

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.unitIso_hom_app_f
code: /-- The unit isomorphism of the equivalence -/
@[simps!]
def unitIso : 𝟭 (Karoubi C) ≅ toKaroubi (Karoubi C) ⋙ inverse C 
goal: ((CategoryTheory.Idempotents.KaroubiKaroubi.unitIso C).hom.app X).f = X.p

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.counitIso_hom_app_f_f
code: /-- The counit isomorphism of the equivalence -/
@[simps]
def counitIso : inverse C ⋙ toKaroubi (Karoubi C) ≅ 𝟭 (Karoubi (Karoubi C)) 
goal: ((CategoryTheory.Idempotents.KaroubiKaroubi.counitIso C).hom.app P).f.f = P.p.f

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.counitIso
code: /-- The counit isomorphism of the equivalence -/
@[simps]
def counitIso : inverse C ⋙ toKaroubi (Karoubi C) ≅ 𝟭 (Karoubi (Karoubi C)) 
goal: (CategoryTheory.Idempotents.KaroubiKaroubi.inverse C).comp
    (CategoryTheory.Idempotents.toKaroubi (CategoryTheory.Idempotents.Karoubi C)) ≅
  CategoryTheory.Functor.id (CategoryTheory.Idempotents.Karoubi (CategoryTheory.Idempotents.Karoubi C))

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.counitIso_inv_app_f_f
code: /-- The counit isomorphism of the equivalence -/
@[simps]
def counitIso : inverse C ⋙ toKaroubi (Karoubi C) ≅ 𝟭 (Karoubi (Karoubi C)) 
goal: ((CategoryTheory.Idempotents.KaroubiKaroubi.counitIso C).inv.app P).f.f = P.p.f

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

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.equivalence_functor
code: /-- The equivalence `Karoubi C ≌ Karoubi (Karoubi C)` -/
@[simps]
def equivalence : Karoubi C ≌ Karoubi (Karoubi C) 
goal: (CategoryTheory.Idempotents.KaroubiKaroubi.equivalence C).functor =
  CategoryTheory.Idempotents.toKaroubi (CategoryTheory.Idempotents.Karoubi C)

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.equivalence_inverse
code: /-- The equivalence `Karoubi C ≌ Karoubi (Karoubi C)` -/
@[simps]
def equivalence : Karoubi C ≌ Karoubi (Karoubi C) 
goal: (CategoryTheory.Idempotents.KaroubiKaroubi.equivalence C).inverse = CategoryTheory.Idempotents.KaroubiKaroubi.inverse C

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.equivalence_counitIso
code: /-- The equivalence `Karoubi C ≌ Karoubi (Karoubi C)` -/
@[simps]
def equivalence : Karoubi C ≌ Karoubi (Karoubi C) 
goal: (CategoryTheory.Idempotents.KaroubiKaroubi.equivalence C).counitIso =
  CategoryTheory.Idempotents.KaroubiKaroubi.counitIso C

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.equivalence_unitIso
code: /-- The equivalence `Karoubi C ≌ Karoubi (Karoubi C)` -/
@[simps]
def equivalence : Karoubi C ≌ Karoubi (Karoubi C) 
goal: (CategoryTheory.Idempotents.KaroubiKaroubi.equivalence C).unitIso = CategoryTheory.Idempotents.KaroubiKaroubi.unitIso C

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.equivalence.additive_functor
code: instance equivalence.additive_functor [Preadditive C] :
  Functor.Additive (equivalence C).functor 
goal: (CategoryTheory.Idempotents.KaroubiKaroubi.equivalence C).functor.Additive

full_name: CategoryTheory.Idempotents.KaroubiKaroubi.equivalence.additive_inverse
code: instance equivalence.additive_inverse [Preadditive C] :
  Functor.Additive (equivalence C).inverse 
goal: (CategoryTheory.Idempotents.KaroubiKaroubi.equivalence C).inverse.Additive