-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.RightDerived (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
inst_2 : Abelian.{v, u} C
inst_3 : HasInjectiveResolutions.{v, u} C
inst_4 : Abelian.{v_1, u_1} D
X : C
I : InjectiveResolution.{v, u} X
F : Functor.{v, v_1, u, u_1} C D
inst_5 : F.Additive
⊢ Eq (α := F.obj X ⟶ (F.rightDerived 0).obj X)
    (CategoryStruct.comp (Y :=
        HomologicalComplex.cycles
          (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
            (F.mapHomologicalComplex (ComplexShape.up ℕ)) (cocomplex (injectiveResolution X)))
          0)
        (toRightDerivedZero' (injectiveResolution X) F :
          F.obj X ⟶
            HomologicalComplex.cycles
              (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
                (F.mapHomologicalComplex (ComplexShape.up ℕ)) (cocomplex (injectiveResolution X)))
              0)
        (HomologicalComplex.homologyπ
              (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
                (F.mapHomologicalComplex (ComplexShape.up ℕ)) (cocomplex (injectiveResolution X)))
              0 ≫
            NatTrans.app (Iso.inv (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0))
              (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
                (F.mapHomologicalComplex (ComplexShape.up ℕ)) (cocomplex (injectiveResolution X))) :
          HomologicalComplex.cycles
              (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
                (F.mapHomologicalComplex (ComplexShape.up ℕ)) (cocomplex (injectiveResolution X)))
              0 ⟶
            (F.rightDerived 0).obj X) :
      F.obj X ⟶ (F.rightDerived 0).obj X)
    (CategoryStruct.comp (Y :=
        HomologicalComplex.cycles (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) 0)
        (I.toRightDerivedZero' F :
          F.obj X ⟶ HomologicalComplex.cycles (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) 0)
        (HomologicalComplex.homologyπ (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) 0 ≫
            NatTrans.app (Iso.inv (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0))
                (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) ≫
              Functor.map (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0)
                (Iso.inv (I.isoRightDerivedToHomotopyCategoryObj F)) :
          HomologicalComplex.cycles (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) 0 ⟶
            (F.rightDerived 0).obj X) :
      F.obj X ⟶ (F.rightDerived 0).obj X)
have h₁ : Eq (α := F.obj X ⟶ HomologicalComplex.cycles (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) 0)
  (F.map (𝟙 X) ≫ I.toRightDerivedZero' F :
    F.obj X ⟶ HomologicalComplex.cycles (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) 0)
  (CategoryStruct.comp (Y :=
      HomologicalComplex.cycles
        (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
          (F.mapHomologicalComplex (ComplexShape.up ℕ)) (cocomplex (injectiveResolution X)))
        0)
      (toRightDerivedZero' (injectiveResolution X) F :
        F.obj X ⟶
          HomologicalComplex.cycles
            (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
              (F.mapHomologicalComplex (ComplexShape.up ℕ)) (cocomplex (injectiveResolution X)))
            0)
      (HomologicalComplex.cyclesMap
          (Functor.map (F.mapHomologicalComplex (ComplexShape.up ℕ)) (desc (𝟙 X) I (injectiveResolution X))) 0 :
        HomologicalComplex.cycles
            (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
              (F.mapHomologicalComplex (ComplexShape.up ℕ)) (cocomplex (injectiveResolution X)))
            0 ⟶
          HomologicalComplex.cycles (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) 0) :
    F.obj X ⟶ HomologicalComplex.cycles (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) 0)
Tactic: have h₁ := InjectiveResolution.toRightDerivedZero'_naturality
    (𝟙 X) (injectiveResolution X) I (desc (𝟙 X) _ _) (by simp) F