-- Mathlib/Algebra/Homology/BifunctorShift.lean
lemma mapBifunctorShift₁Iso_trans_mapBifunctorShift₂Iso :
    mapBifunctorShift₁Iso K₁ (K₂⟦y⟧) F x ≪≫
      (CategoryTheory.shiftFunctor _ x).mapIso (mapBifunctorShift₂Iso K₁ K₂ F y) =
      (x * y).negOnePow • (mapBifunctorShift₂Iso (K₁⟦x⟧) K₂ F y ≪≫
        (CategoryTheory.shiftFunctor _ y).mapIso (mapBifunctorShift₁Iso K₁ K₂ F x) ≪≫
          (shiftFunctorComm (CochainComplex D ℤ) x y).app _)

Proof state:
case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ (HomologicalComplex₂.total.map
          (K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F
              x).hom
          (ComplexShape.up ℤ) ≫
        ((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
                ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂)).totalShift₁Iso
            x).hom) ≫
      (CategoryTheory.shiftFunctor (CochainComplex D ℤ) x).map
        (HomologicalComplex₂.total.map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom (ComplexShape.up ℤ) ≫
          ((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).totalShift₂Iso
              y).hom) =
    (x * y).negOnePow •
      (HomologicalComplex₂.total.map
            (((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
                y).hom
            (ComplexShape.up ℤ) ≫
          ((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj
                      ((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁)).obj
                  K₂).totalShift₂Iso
              y).hom) ≫
        (CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).map
            (HomologicalComplex₂.total.map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom (ComplexShape.up ℤ) ≫
              ((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
                      K₂).totalShift₁Iso
                  x).hom) ≫
          (shiftFunctorComm (CochainComplex D ℤ) x y).hom.app (K₁.mapBifunctor K₂ F)

Tactic:
rw [Functor.map_comp, Functor.map_comp, assoc, assoc, assoc,
  ← HomologicalComplex₂.totalShift₁Iso_hom_naturality_assoc,
  HomologicalComplex₂.totalShift₁Iso_hom_totalShift₂Iso_hom,
  ← HomologicalComplex₂.totalShift₂Iso_hom_naturality_assoc,
  Linear.comp_units_smul, Linear.comp_units_smul,
  smul_left_cancel_iff,
  ← HomologicalComplex₂.total.map_comp_assoc,
  ← HomologicalComplex₂.total.map_comp_assoc,
  ← HomologicalComplex₂.total.map_comp_assoc]

Result:
case w
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
⊢ HomologicalComplex₂.total.map
        (((K₁.mapBifunctorHomologicalComplexShift₁Iso ((CategoryTheory.shiftFunctor (CochainComplex C₂ ℤ) y).obj K₂) F
                x).hom ≫
            (HomologicalComplex₂.shiftFunctor₁ D x).map (K₁.mapBifunctorHomologicalComplexShift₂Iso K₂ F y).hom) ≫
          (HomologicalComplex₂.shiftFunctor₁₂CommIso D x y).hom.app
            (((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂))
        (ComplexShape.up ℤ) ≫
      (((HomologicalComplex₂.shiftFunctor₁ D x).obj
                (((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
                  K₂)).totalShift₂Iso
            y).hom ≫
        (CategoryTheory.shiftFunctor (HomologicalComplex D (ComplexShape.up ℤ)) y).map
            ((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).totalShift₁Iso
                x).hom ≫
          (shiftFunctorComm (CochainComplex D ℤ) x y).hom.app
            ((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).total
              (ComplexShape.up ℤ)) =
    HomologicalComplex₂.total.map
        ((((CategoryTheory.shiftFunctor (CochainComplex C₁ ℤ) x).obj K₁).mapBifunctorHomologicalComplexShift₂Iso K₂ F
              y).hom ≫
          (HomologicalComplex₂.shiftFunctor₂ D y).map (K₁.mapBifunctorHomologicalComplexShift₁Iso K₂ F x).hom)
        (ComplexShape.up ℤ) ≫
      (((HomologicalComplex₂.shiftFunctor₁ D x).obj
                (((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj
                  K₂)).totalShift₂Iso
            y).hom ≫
        (CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).map
            ((((F.mapBifunctorHomologicalComplex (ComplexShape.up ℤ) (ComplexShape.up ℤ)).obj K₁).obj K₂).totalShift₁Iso
                x).hom ≫
          (shiftFunctorComm (CochainComplex D ℤ) x y).hom.app (K₁.mapBifunctor K₂ F)

Proof state:
case w.e_a.e_φ.h.h
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁸ : Category.{u_5, u_1} C₁
inst✝⁷ : Category.{u_6, u_2} C₂
inst✝⁶ : Category.{u_4, u_3} D
inst✝⁵ : Preadditive C₁
inst✝⁴ : Preadditive C₂
inst✝³ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ : CochainComplex C₂ ℤ
F : C₁ ⥤ C₂ ⥤ D
inst✝² : F.Additive
inst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
x y : ℤ
inst✝ : K₁.HasMapBifunctor K₂ F
a b : ℤ
⊢ (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:
simp only [id_comp]

Result:
no goals