-- 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 f g
n : ℤ
simp :
  HAdd.hAdd (α := op (L.X (-n)) ⟶ op (K.X (-n))) (β := op (L.X (-n)) ⟶ op (K.X (-n)))
      (Quiver.Hom.op ((dNext (-n) : (a : (i j : ℤ) → K.X i ⟶ L.X j) → K.X (-n) ⟶ L.X (-n)) h.hom) :
        op (L.X (-n)) ⟶ op (K.X (-n)))
      (Quiver.Hom.op ((prevD (-n) : (a : (i j : ℤ) → K.X i ⟶ L.X j) → K.X (-n) ⟶ L.X (-n)) h.hom) :
        op (L.X (-n)) ⟶ op (K.X (-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)))
          fun (p q : ℤ) ↦ Quiver.Hom.op (h.hom (-q) (-p)) :
        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)))
          fun (p q : ℤ) ↦ Quiver.Hom.op (h.hom (-q) (-p)) :
        op (L.X (-n)) ⟶ op (K.X (-n)))
⊢ 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)))
            fun (p q : ℤ) ↦ Quiver.Hom.op (h.hom (-q) (-p)) :
          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)))
            fun (p q : ℤ) ↦ Quiver.Hom.op (h.hom (-q) (-p)) :
          op (L.X (-n)) ⟶ op (K.X (-n))) +
      Quiver.Hom.op (g.f (-n))

Tactic:
simp only [h.comm, op_add, add_left_inj]

Premises used: add_left_inj, CategoryTheory.op_add, Homotopy.comm