-- Module: Mathlib.Algebra.Homology.CochainComplexOpposite
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
K L : CochainComplex.{v_1, u_1, 0} C ℤ
f g : K ⟶ L
h :
  Homotopy.{v_1, u_1, 0} (ι := ℤ) (V := Cᵒᵖ) (c := ComplexShape.up ℤ) (C :=
    Functor.obj (Equivalence.functor (opEquivalence C)) (op L)) (D :=
    Functor.obj (Equivalence.functor (opEquivalence C)) (op K))
    (CategoryTheory.Functor.map (Equivalence.functor (opEquivalence C)) f.op)
    (CategoryTheory.Functor.map (Equivalence.functor (opEquivalence C)) g.op)
H :
  ∀ (p q p' q' : ℤ) (hp : p = p') (hq : q = q'),
    h.hom p q =
      Quiver.Hom.op (Iso.hom (HomologicalComplex.XIsoOfEq L ⋯)) ≫
        h.hom p' q' ≫ Quiver.Hom.op (Iso.hom (HomologicalComplex.XIsoOfEq K ⋯))
n : ℤ
this :
  Quiver.Hom.op (f.f (-n)) =
    HAdd.hAdd (α := op (L.X (-n)) ⟶ op (K.X (-n))) (β := op (L.X (-n)) ⟶ op (K.X (-n)))
        (((dNext.{v_1, u_1, 0} (V := Cᵒᵖ) (c := ComplexShape.up ℤ) (C :=
                  (HomologicalComplex.op L).restriction ComplexShape.embeddingUpIntDownInt) (D :=
                  (HomologicalComplex.op K).restriction ComplexShape.embeddingUpIntDownInt) n :
                ((i j : ℤ) →
                    ((HomologicalComplex.op L).restriction ComplexShape.embeddingUpIntDownInt).X i ⟶
                      ((HomologicalComplex.op K).restriction ComplexShape.embeddingUpIntDownInt).X j) →+
                  (((HomologicalComplex.op L).restriction ComplexShape.embeddingUpIntDownInt).X n ⟶
                    ((HomologicalComplex.op K).restriction ComplexShape.embeddingUpIntDownInt).X n)) :
              (a : (i j : ℤ) → op (L.X (-i)) ⟶ op (K.X (-j))) → op (L.X (-n)) ⟶ op (K.X (-n)))
            h.hom :
          op (L.X (-n)) ⟶ op (K.X (-n)))
        (((prevD.{v_1, u_1, 0} (V := Cᵒᵖ) (c := ComplexShape.up ℤ) (C :=
                  (HomologicalComplex.op L).restriction ComplexShape.embeddingUpIntDownInt) (D :=
                  (HomologicalComplex.op K).restriction ComplexShape.embeddingUpIntDownInt) n :
                ((i j : ℤ) →
                    ((HomologicalComplex.op L).restriction ComplexShape.embeddingUpIntDownInt).X i ⟶
                      ((HomologicalComplex.op K).restriction ComplexShape.embeddingUpIntDownInt).X j) →+
                  (((HomologicalComplex.op L).restriction ComplexShape.embeddingUpIntDownInt).X n ⟶
                    ((HomologicalComplex.op K).restriction ComplexShape.embeddingUpIntDownInt).X n)) :
              (a : (i j : ℤ) → op (L.X (-i)) ⟶ op (K.X (-j))) → op (L.X (-n)) ⟶ op (K.X (-n)))
            h.hom :
          op (L.X (-n)) ⟶ op (K.X (-n))) +
      Quiver.Hom.op (g.f (-n))
⊢ Eq (α := op (L.X (-n)) ⟶ op (K.X (-n)))
    (CategoryStruct.comp (Y := op (L.X (-(n + 1)))) (Quiver.Hom.op (L.d (-(n + 1)) (-n))) (h.hom (n + 1) n))
    (CategoryStruct.comp (Y := op (L.X (-(n + 1)))) (Quiver.Hom.op (L.d (-(n + 1)) (-n)))
      ((Quiver.Hom.op (Iso.hom (HomologicalComplex.XIsoOfEq L ⋯)) ≫ h.hom (- -(n + 1)) (- -n)) ≫
        Quiver.Hom.op (Iso.hom (HomologicalComplex.XIsoOfEq K ⋯))))

Tactic:
simp [H (- -(n + 1)) (- -n) (n + 1) n (by simp) (by simp), ← op_comp_assoc, ← op_comp]

Premises used: CategoryTheory.op_comp, CategoryTheory.op_comp_assoc, HAdd.hAdd, OfNat.ofNat, Neg.neg