-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.RightDerived (goal)
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 Y : C
f : X ⟶ Y
I : InjectiveResolution.{v, u} X
J : InjectiveResolution.{v, u} Y
φ : I.cocomplex ⟶ J.cocomplex
comm :
  Eq (α := HomologicalComplex.X.{v, u, 0} (Functor.obj (CochainComplex.single₀ C) X) 0 ⟶ J.cocomplex.X 0)
    (I.ι.f 0 ≫ φ.f 0) (f ≫ J.ι.f 0)
F : Functor.{v, v_1, u, u_1} C D
inst_5 : F.Additive
⊢ Eq (α :=
    Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.up ℕ))
      (Functor.obj F.rightDerivedToHomotopyCategory X)
      (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ))
        J.cocomplex))
    (Functor.map F.rightDerivedToHomotopyCategory f ≫ Iso.hom (J.isoRightDerivedToHomotopyCategoryObj F))
    (Iso.hom (I.isoRightDerivedToHomotopyCategoryObj F) ≫
      Functor.map (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)) φ)
have dsimp : Eq (α :=
  Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.up ℕ))
    (Functor.obj (HomotopyCategory.quotient D (ComplexShape.up ℕ))
      (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
        (F.mapHomologicalComplex (ComplexShape.up ℕ))
        (Quotient.as (r := homotopic C (ComplexShape.up ℕ)) (Functor.obj (injectiveResolutions C) X))))
    (Functor.obj (HomotopyCategory.quotient D (ComplexShape.up ℕ))
      (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) J.cocomplex)))
  (CategoryStruct.comp (Y :=
      Functor.obj (HomotopyCategory.quotient D (ComplexShape.up ℕ))
        (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
          (F.mapHomologicalComplex (ComplexShape.up ℕ))
          (Quotient.as (r := homotopic C (ComplexShape.up ℕ)) (Functor.obj (injectiveResolutions C) Y))))
      (Functor.map (F.mapHomotopyCategory (ComplexShape.up ℕ)) (Functor.map (injectiveResolutions C) f) :
        Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.up ℕ))
          (Functor.obj (C := HomotopyCategory.{v, u, 0} C (ComplexShape.up ℕ))
            (F.mapHomotopyCategory (ComplexShape.up ℕ)) (Functor.obj (injectiveResolutions C) X))
          (Functor.obj (C := HomotopyCategory.{v, u, 0} C (ComplexShape.up ℕ))
            (F.mapHomotopyCategory (ComplexShape.up ℕ)) (Functor.obj (injectiveResolutions C) Y)))
      (CategoryStruct.comp (Y :=
        Functor.obj (HomotopyCategory.quotient D (ComplexShape.up ℕ))
          (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ))
            ((Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℕ)) J.cocomplex).as (r :=
              homotopic C (ComplexShape.up ℕ)))))
        (Functor.map (F.mapHomotopyCategory (ComplexShape.up ℕ)) (Iso.hom J.iso) :
          Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.up ℕ))
            (Functor.obj (C := HomotopyCategory.{v, u, 0} C (ComplexShape.up ℕ))
              (F.mapHomotopyCategory (ComplexShape.up ℕ)) (Functor.obj (injectiveResolutions C) Y))
            (Functor.obj (F.mapHomotopyCategory (ComplexShape.up ℕ))
              (Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℕ)) J.cocomplex)))
        (NatTrans.app (Iso.hom (F.mapHomotopyCategoryFactors (ComplexShape.up ℕ))) J.cocomplex :
          Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.up ℕ))
            (Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℕ) ⋙ F.mapHomotopyCategory (ComplexShape.up ℕ))
              J.cocomplex)
            (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ))
              J.cocomplex))) :
    Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.up ℕ))
      (Functor.obj (HomotopyCategory.quotient D (ComplexShape.up ℕ))
        (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
          (F.mapHomologicalComplex (ComplexShape.up ℕ))
          (Quotient.as (r := homotopic C (ComplexShape.up ℕ)) (Functor.obj (injectiveResolutions C) X))))
      (Functor.obj (HomotopyCategory.quotient D (ComplexShape.up ℕ))
        (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) J.cocomplex)))
  (CategoryStruct.comp (Y :=
        Functor.obj (HomotopyCategory.quotient D (ComplexShape.up ℕ))
          (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ))
            ((Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℕ)) I.cocomplex).as (r :=
              homotopic C (ComplexShape.up ℕ)))))
        (Functor.map (F.mapHomotopyCategory (ComplexShape.up ℕ)) (Iso.hom I.iso) :
          Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.up ℕ))
            (Functor.obj (C := HomotopyCategory.{v, u, 0} C (ComplexShape.up ℕ))
              (F.mapHomotopyCategory (ComplexShape.up ℕ)) (Functor.obj (injectiveResolutions C) X))
            (Functor.obj (F.mapHomotopyCategory (ComplexShape.up ℕ))
              (Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℕ)) I.cocomplex)))
        (NatTrans.app (Iso.hom (F.mapHomotopyCategoryFactors (ComplexShape.up ℕ))) I.cocomplex :
          Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.up ℕ))
            (Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℕ) ⋙ F.mapHomotopyCategory (ComplexShape.up ℕ))
              I.cocomplex)
            (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ))
              I.cocomplex)) ≫
      Functor.map (HomotopyCategory.quotient D (ComplexShape.up ℕ))
        (Functor.map (F.mapHomologicalComplex (ComplexShape.up ℕ)) φ) :
    Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.up ℕ))
      (Functor.obj (HomotopyCategory.quotient D (ComplexShape.up ℕ))
        (Functor.obj (C := HomologicalComplex.{v, u, 0} C (ComplexShape.up ℕ))
          (F.mapHomologicalComplex (ComplexShape.up ℕ))
          (Quotient.as (r := homotopic C (ComplexShape.up ℕ)) (Functor.obj (injectiveResolutions C) X))))
      (Functor.obj (HomotopyCategory.quotient D (ComplexShape.up ℕ))
        (Functor.obj (F.mapHomologicalComplex (ComplexShape.up ℕ)) J.cocomplex)))
Tactic: dsimp [Functor.rightDerivedToHomotopyCategory, isoRightDerivedToHomotopyCategoryObj]