-- Lean 4 proof step from Mathlib.Algebra.Homology.BifunctorShift (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_3} D
inst_3 : Preadditive.{v_1, u_1} C₁
inst_4 : Preadditive.{v_2, u_2} C₂
inst_5 : Preadditive.{v_3, u_3} D
K₁ : CochainComplex.{v_1, u_1, 0} C₁ ℤ
K₂ : CochainComplex.{v_2, u_2, 0} C₂ ℤ
F : C₁ ⥤ CategoryTheory.Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : F.Additive
inst_7 : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst_8 : K₁.HasMapBifunctor K₂ F
a b : ℤ
⊢ Eq (α :=
    X
        (X
          (Functor.obj
            (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ))
              (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_1, u_1, 0} C₁ (ComplexShape.up ℤ)) x)
                K₁))
            (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_2, u_2, 0} C₂ ℤ) y) K₂))
          a)
        b ⟶
      X
        (X
          (Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x ⋙ HomologicalComplex₂.shiftFunctor₂ D y)
            (Functor.obj (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
              K₂))
          a)
        b)
    (Hom.f
        (Hom.f
          (CategoryStruct.comp (Y :=
            Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x)
              (Functor.obj (HomologicalComplex₂.shiftFunctor₂ D y)
                (Functor.obj (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
                  K₂)))
            (CategoryStruct.comp (Y :=
                Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x)
                  (Functor.obj
                    (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
                    (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_2, u_2, 0} C₂ ℤ) y) K₂)))
                (Iso.hom
                    (K₁.mapBifunctorHomologicalComplexShift₁Iso
                      (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_2, u_2, 0} C₂ ℤ) y) K₂) F x) :
                  Quiver.Hom (V := HomologicalComplex₂.{v_3, u_3, 0, 0} D (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                    (Functor.obj
                      (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                        (Functor.obj
                          (CategoryTheory.shiftFunctor (HomologicalComplex.{v_1, u_1, 0} C₁ (ComplexShape.up ℤ)) x) K₁))
                      (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_2, u_2, 0} C₂ ℤ) y) K₂))
                    (Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x)
                      (Functor.obj
                        (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
                        (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_2, u_2, 0} C₂ ℤ) y) K₂))))
                (CategoryTheory.Functor.map (HomologicalComplex₂.shiftFunctor₁ D x)
                    (Iso.hom (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y)) :
                  Quiver.Hom (V := HomologicalComplex₂.{v_3, u_3, 0, 0} D (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                    (Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x)
                      (Functor.obj
                        (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
                        (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_2, u_2, 0} C₂ ℤ) y) K₂)))
                    (Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x)
                      (Functor.obj (HomologicalComplex₂.shiftFunctor₂ D y)
                        (Functor.obj
                          (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
                          K₂)))) :
              Quiver.Hom (V := HomologicalComplex₂.{v_3, u_3, 0, 0} D (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                (Functor.obj
                  (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                    (Functor.obj
                      (CategoryTheory.shiftFunctor (HomologicalComplex.{v_1, u_1, 0} C₁ (ComplexShape.up ℤ)) x) K₁))
                  (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_2, u_2, 0} C₂ ℤ) y) K₂))
                (Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x)
                  (Functor.obj (HomologicalComplex₂.shiftFunctor₂ D y)
                    (Functor.obj
                      (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁) K₂))))
            (NatTrans.app (Iso.hom (HomologicalComplex₂.shiftFunctor₁₂CommIso D x y))
                (Functor.obj (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
                  K₂) :
              Quiver.Hom (V := HomologicalComplex₂.{v_3, u_3, 0, 0} D (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                (Functor.obj (HomologicalComplex₂.shiftFunctor₂ D y ⋙ HomologicalComplex₂.shiftFunctor₁ D x)
                  (Functor.obj
                    (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁) K₂))
                (Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x ⋙ HomologicalComplex₂.shiftFunctor₂ D y)
                  (Functor.obj
                    (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁) K₂))))
          a)
        b :
      X
          (X
            (Functor.obj
              (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_1, u_1, 0} C₁ (ComplexShape.up ℤ)) x)
                  K₁))
              (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_2, u_2, 0} C₂ ℤ) y) K₂))
            a)
          b ⟶
        X
          (X
            (Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x ⋙ HomologicalComplex₂.shiftFunctor₂ D y)
              (Functor.obj (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
                K₂))
            a)
          b)
    (Hom.f
        (Hom.f
          (CategoryStruct.comp (Y :=
            Functor.obj (HomologicalComplex₂.shiftFunctor₂ D y)
              (Functor.obj
                (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                  (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C₁ ℤ) x) K₁))
                K₂))
            (Iso.hom
                ((Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C₁ ℤ) x)
                      K₁).mapBifunctorHomologicalComplexShift₂Iso
                  K₂ F y) :
              Quiver.Hom (V := HomologicalComplex₂.{v_3, u_3, 0, 0} D (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                (Functor.obj
                  (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                    (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C₁ ℤ) x) K₁))
                  (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_2, u_2, 0} C₂ (ComplexShape.up ℤ)) y)
                    K₂))
                (Functor.obj (HomologicalComplex₂.shiftFunctor₂ D y)
                  (Functor.obj
                    (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                      (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C₁ ℤ) x) K₁))
                    K₂)))
            (CategoryTheory.Functor.map (HomologicalComplex₂.shiftFunctor₂ D y)
                (Iso.hom (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x)) :
              Quiver.Hom (V := HomologicalComplex₂.{v_3, u_3, 0, 0} D (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                (Functor.obj (HomologicalComplex₂.shiftFunctor₂ D y)
                  (Functor.obj
                    (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                      (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C₁ ℤ) x) K₁))
                    K₂))
                (Functor.obj (HomologicalComplex₂.shiftFunctor₂ D y)
                  (Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x)
                    (Functor.obj
                      (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
                      K₂)))))
          a)
        b :
      X
          (X
            (Functor.obj
              (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ))
                (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v_1, u_1, 0} C₁ (ComplexShape.up ℤ)) x)
                  K₁))
              (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_2, u_2, 0} C₂ ℤ) y) K₂))
            a)
          b ⟶
        X
          (X
            (Functor.obj (HomologicalComplex₂.shiftFunctor₁ D x ⋙ HomologicalComplex₂.shiftFunctor₂ D y)
              (Functor.obj (Functor.obj (F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)) K₁)
                K₂))
            a)
          b)
have w : Eq.{v_3 + 1} (α := (F.obj (K₁.X (a + x))).obj (K₂.X (b + y)) ⟶ (F.obj (K₁.X (a + x))).obj (K₂.X (b + y)))
  ((𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))) ≫ 𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y)))) ≫
    𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))))
  (𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))) ≫ 𝟙 ((F.obj (K₁.X (a + x))).obj (K₂.X (b + y))))
Tactic: dsimp [HomologicalComplex₂.shiftFunctor₁₂CommIso]