-- Module: Mathlib.CategoryTheory.Abelian.LeftDerived
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 : HasProjectiveResolutions.{v, u} C
inst_4 : Abelian.{v_1, u_1} D
F G : Functor.{v, v_1, u, u_1} C D
inst_5 : F.Additive
inst_6 : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution.{v, u} X
dsimp :
  Eq (α :=
    Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.down ℕ))
      (Functor.obj F.leftDerivedToHomotopyCategory X)
      (Functor.obj (HomotopyCategory.quotient D (ComplexShape.down ℕ))
        (Functor.obj (G.mapHomologicalComplex (ComplexShape.down ℕ)) P.complex)))
    (CategoryStruct.comp (Y := Functor.obj G.leftDerivedToHomotopyCategory X)
        (Functor.map (HomotopyCategory.quotient D (ComplexShape.down ℕ))
          (NatTrans.app (NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ))
            (Quotient.as (r := homotopic C (ComplexShape.down ℕ)) (Functor.obj (projectiveResolutions C) X))))
        (Functor.map (G.mapHomotopyCategory (ComplexShape.down ℕ)) (Iso.hom P.iso) ≫
          𝟙
            (Functor.obj (HomotopyCategory.quotient D (ComplexShape.down ℕ))
              (Functor.obj (G.mapHomologicalComplex (ComplexShape.down ℕ)) P.complex))) :
      Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.down ℕ))
        (Functor.obj F.leftDerivedToHomotopyCategory X)
        (Functor.obj (HomotopyCategory.quotient D (ComplexShape.down ℕ))
          (Functor.obj (G.mapHomologicalComplex (ComplexShape.down ℕ)) P.complex)))
    (CategoryStruct.comp (Y :=
        Functor.obj (HomotopyCategory.quotient D (ComplexShape.down ℕ))
          (Functor.obj (F.mapHomologicalComplex (ComplexShape.down ℕ)) P.complex))
        (Functor.map (F.mapHomotopyCategory (ComplexShape.down ℕ)) (Iso.hom P.iso) ≫
          𝟙
            (Functor.obj (HomotopyCategory.quotient D (ComplexShape.down ℕ))
              (Functor.obj (F.mapHomologicalComplex (ComplexShape.down ℕ)) P.complex)))
        (Functor.map (HomotopyCategory.quotient D (ComplexShape.down ℕ))
          (NatTrans.app (NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)) P.complex)) :
      Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.down ℕ))
        (Functor.obj F.leftDerivedToHomotopyCategory X)
        (Functor.obj (HomotopyCategory.quotient D (ComplexShape.down ℕ))
          (Functor.obj (G.mapHomologicalComplex (ComplexShape.down ℕ)) P.complex)))
⊢ Eq (α :=
    Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.down ℕ))
      (Functor.obj F.leftDerivedToHomotopyCategory X)
      (Functor.obj (G.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.down ℕ))
        P.complex))
    (CategoryStruct.comp (Y := Functor.obj G.leftDerivedToHomotopyCategory X)
        (NatTrans.app (NatTrans.leftDerivedToHomotopyCategory α) X)
        (Iso.hom (P.isoLeftDerivedToHomotopyCategoryObj G)) :
      Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.down ℕ))
        (Functor.obj F.leftDerivedToHomotopyCategory X)
        (Functor.obj (G.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.down ℕ))
          P.complex))
    (Iso.hom (P.isoLeftDerivedToHomotopyCategoryObj F) ≫
        Functor.map (HomotopyCategory.quotient D (ComplexShape.down ℕ))
          (NatTrans.app (NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)) P.complex) :
      Quiver.Hom (V := HomotopyCategory.{v_1, u_1, 0} D (ComplexShape.down ℕ))
        (Functor.obj F.leftDerivedToHomotopyCategory X)
        (Functor.obj (G.mapHomologicalComplex (ComplexShape.down ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.down ℕ))
          P.complex))

Tactic:
dsimp [isoLeftDerivedToHomotopyCategoryObj, Functor.mapHomotopyCategoryFactors,
    NatTrans.leftDerivedToHomotopyCategory]

Premises used: CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj, CategoryTheory.NatTrans.leftDerivedToHomotopyCategory, CategoryTheory.Functor.mapHomotopyCategoryFactors