Theorem: CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor_map_hom₂
File: Mathlib/CategoryTheory/Triangulated/Opposite.lean

Proof steps:
state_before: C : Type u_1
inst✝¹ : Category.{?u.90051, u_1} C
inst✝ : HasShift C ℤ
T₁ T₂ : (Triangle C)ᵒᵖ
φ : T₁ ⟶ T₂
⊢ ((fun T =>
            Triangle.mk (Opposite.unop T).mor₂.op (Opposite.unop T).mor₁.op
              ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T).obj₁) ≫
                (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T).mor₃.op))
          T₁).mor₃ ≫
      (shiftFunctor Cᵒᵖ 1).map φ.unop.hom₃.op =
    φ.unop.hom₁.op ≫
      ((fun T =>
            Triangle.mk (Opposite.unop T).mor₂.op (Opposite.unop T).mor₁.op
              ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T).obj₁) ≫
                (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T).mor₃.op))
          T₂).mor₃
state_after: C : Type u_1
inst✝¹ : Category.{?u.90051, u_1} C
inst✝ : HasShift C ℤ
T₁ T₂ : (Triangle C)ᵒᵖ
φ : T₁ ⟶ T₂
⊢ ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T₁).obj₁) ≫
        (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T₁).mor₃.op) ≫
      (shiftFunctor Cᵒᵖ 1).map φ.unop.hom₃.op =
    φ.unop.hom₁.op ≫
      (opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T₂).obj₁) ≫
        (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T₂).mor₃.op
tactic: dsimp
state_before: C : Type u_1
inst✝¹ : Category.{?u.90051, u_1} C
inst✝ : HasShift C ℤ
T₁ T₂ : (Triangle C)ᵒᵖ
φ : T₁ ⟶ T₂
⊢ ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T₁).obj₁) ≫
        (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T₁).mor₃.op) ≫
      (shiftFunctor Cᵒᵖ 1).map φ.unop.hom₃.op =
    φ.unop.hom₁.op ≫
      (opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T₂).obj₁) ≫
        (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T₂).mor₃.op
state_after: C : Type u_1
inst✝¹ : Category.{?u.90051, u_1} C
inst✝ : HasShift C ℤ
T₁ T₂ : (Triangle C)ᵒᵖ
φ : T₁ ⟶ T₂
⊢ (opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T₁).obj₁) ≫
      (shiftFunctor Cᵒᵖ 1).map ((shiftFunctor C 1).map φ.unop.hom₁).op ≫
        (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T₂).mor₃.op =
    (opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T₁).obj₁) ≫
      (shiftFunctor Cᵒᵖ 1).map ((shiftFunctor C 1).map φ.unop.hom₁.op.unop).op ≫
        (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T₂).mor₃.op
tactic: rw [assoc, ← Functor.map_comp, ← op_comp, ← φ.unop.comm₃, op_comp, Functor.map_comp,
          opShiftFunctorEquivalence_counitIso_inv_naturality_assoc]
state_before: C : Type u_1
inst✝¹ : Category.{?u.90051, u_1} C
inst✝ : HasShift C ℤ
T₁ T₂ : (Triangle C)ᵒᵖ
φ : T₁ ⟶ T₂
⊢ (opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T₁).obj₁) ≫
      (shiftFunctor Cᵒᵖ 1).map ((shiftFunctor C 1).map φ.unop.hom₁).op ≫
        (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T₂).mor₃.op =
    (opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op (Opposite.unop T₁).obj₁) ≫
      (shiftFunctor Cᵒᵖ 1).map ((shiftFunctor C 1).map φ.unop.hom₁.op.unop).op ≫
        (shiftFunctor Cᵒᵖ 1).map (Opposite.unop T₂).mor₃.op
state_after: no goals
tactic: rfl