∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
  [inst_2 : CategoryTheory.HasInjectiveResolutions C] {X Y : C} (f : X ⟶ Y) (I : CategoryTheory.InjectiveResolution X)
  (J : CategoryTheory.InjectiveResolution Y) (φ : I.cocomplex ⟶ J.cocomplex),
  CategoryTheory.CategoryStruct.comp (I.ι.f 0) (φ.f 0) = CategoryTheory.CategoryStruct.comp f (J.ι.f 0) →
    CategoryTheory.CategoryStruct.comp I.iso.inv ((CategoryTheory.injectiveResolutions C).map f) =
      CategoryTheory.CategoryStruct.comp ((HomotopyCategory.quotient C (ComplexShape.up ℕ)).map φ) J.iso.inv