-- Module: Mathlib.CategoryTheory.Abelian.RightDerived
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
dsimp :
  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)
⊢ Eq (α := F.obj X ⟶ (F.rightDerived 0).obj X) (NatTrans.app F.toRightDerivedZero 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)
      (Iso.hom (CochainComplex.isoHomologyπ₀ (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex)) ≫
          Iso.inv (I.isoRightDerivedObj F 0) :
        HomologicalComplex.cycles (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) I.cocomplex) 0 ⟶
          (F.rightDerived 0).obj X))

Tactic:
dsimp [Functor.toRightDerivedZero, isoRightDerivedObj]

Premises used: CategoryTheory.InjectiveResolution.isoRightDerivedObj, CategoryTheory.Functor.toRightDerivedZero