Module: Mathlib.CategoryTheory.Triangulated.Basic

Goal state before:
pp: case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
A B : J ⥤ Triangle C
hom₁ : A ⋙ π₁ ⟶ B ⋙ π₁
hom₂ : A ⋙ π₂ ⟶ B ⋙ π₂
hom₃ : A ⋙ π₃ ⟶ B ⋙ π₃
comm₁ : A.whiskerLeft π₁Toπ₂ ≫ hom₂ = hom₁ ≫ B.whiskerLeft π₁Toπ₂
comm₂ : A.whiskerLeft π₂Toπ₃ ≫ hom₃ = hom₂ ≫ B.whiskerLeft π₂Toπ₃
comm₃ : A.whiskerLeft π₃Toπ₁ ≫ whiskerRight hom₁ (shiftFunctor C 1) = hom₃ ≫ B.whiskerLeft π₃Toπ₁
x✝¹ x✝ : J
φ : x✝¹ ⟶ x✝
⊢ (A.map φ ≫
        { hom₁ := hom₁.app x✝, hom₂ := hom₂.app x✝, hom₃ := hom₃.app x✝, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }).hom₁ =
    ({ hom₁ := hom₁.app x✝¹, hom₂ := hom₂.app x✝¹, hom₃ := hom₃.app x✝¹, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ } ≫
        B.map φ).hom₁

pp: case h₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
A B : J ⥤ Triangle C
hom₁ : A ⋙ π₁ ⟶ B ⋙ π₁
hom₂ : A ⋙ π₂ ⟶ B ⋙ π₂
hom₃ : A ⋙ π₃ ⟶ B ⋙ π₃
comm₁ : A.whiskerLeft π₁Toπ₂ ≫ hom₂ = hom₁ ≫ B.whiskerLeft π₁Toπ₂
comm₂ : A.whiskerLeft π₂Toπ₃ ≫ hom₃ = hom₂ ≫ B.whiskerLeft π₂Toπ₃
comm₃ : A.whiskerLeft π₃Toπ₁ ≫ whiskerRight hom₁ (shiftFunctor C 1) = hom₃ ≫ B.whiskerLeft π₃Toπ₁
x✝¹ x✝ : J
φ : x✝¹ ⟶ x✝
⊢ (A.map φ ≫
        { hom₁ := hom₁.app x✝, hom₂ := hom₂.app x✝, hom₃ := hom₃.app x✝, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }).hom₂ =
    ({ hom₁ := hom₁.app x✝¹, hom₂ := hom₂.app x✝¹, hom₃ := hom₃.app x✝¹, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ } ≫
        B.map φ).hom₂

pp: case h₃
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
A B : J ⥤ Triangle C
hom₁ : A ⋙ π₁ ⟶ B ⋙ π₁
hom₂ : A ⋙ π₂ ⟶ B ⋙ π₂
hom₃ : A ⋙ π₃ ⟶ B ⋙ π₃
comm₁ : A.whiskerLeft π₁Toπ₂ ≫ hom₂ = hom₁ ≫ B.whiskerLeft π₁Toπ₂
comm₂ : A.whiskerLeft π₂Toπ₃ ≫ hom₃ = hom₂ ≫ B.whiskerLeft π₂Toπ₃
comm₃ : A.whiskerLeft π₃Toπ₁ ≫ whiskerRight hom₁ (shiftFunctor C 1) = hom₃ ≫ B.whiskerLeft π₃Toπ₁
x✝¹ x✝ : J
φ : x✝¹ ⟶ x✝
⊢ (A.map φ ≫
        { hom₁ := hom₁.app x✝, hom₂ := hom₂.app x✝, hom₃ := hom₃.app x✝, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }).hom₃ =
    ({ hom₁ := hom₁.app x✝¹, hom₂ := hom₂.app x✝¹, hom₃ := hom₃.app x✝¹, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ } ≫
        B.map φ).hom₃

Tactic:
· exact hom₁.naturality φ

Goal state after:
case h₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
A B : J ⥤ Triangle C
hom₁ : A ⋙ π₁ ⟶ B ⋙ π₁
hom₂ : A ⋙ π₂ ⟶ B ⋙ π₂
hom₃ : A ⋙ π₃ ⟶ B ⋙ π₃
comm₁ : A.whiskerLeft π₁Toπ₂ ≫ hom₂ = hom₁ ≫ B.whiskerLeft π₁Toπ₂
comm₂ : A.whiskerLeft π₂Toπ₃ ≫ hom₃ = hom₂ ≫ B.whiskerLeft π₂Toπ₃
comm₃ : A.whiskerLeft π₃Toπ₁ ≫ whiskerRight hom₁ (shiftFunctor C 1) = hom₃ ≫ B.whiskerLeft π₃Toπ₁
x✝¹ x✝ : J
φ : x✝¹ ⟶ x✝
⊢ (A.map φ ≫
        { hom₁ := hom₁.app x✝, hom₂ := hom₂.app x✝, hom₃ := hom₃.app x✝, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }).hom₂ =
    ({ hom₁ := hom₁.app x✝¹, hom₂ := hom₂.app x✝¹, hom₃ := hom₃.app x✝¹, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ } ≫
        B.map φ).hom₂

case h₃
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
J : Type u_1
inst✝ : Category.{v_1, u_1} J
A B : J ⥤ Triangle C
hom₁ : A ⋙ π₁ ⟶ B ⋙ π₁
hom₂ : A ⋙ π₂ ⟶ B ⋙ π₂
hom₃ : A ⋙ π₃ ⟶ B ⋙ π₃
comm₁ : A.whiskerLeft π₁Toπ₂ ≫ hom₂ = hom₁ ≫ B.whiskerLeft π₁Toπ₂
comm₂ : A.whiskerLeft π₂Toπ₃ ≫ hom₃ = hom₂ ≫ B.whiskerLeft π₂Toπ₃
comm₃ : A.whiskerLeft π₃Toπ₁ ≫ whiskerRight hom₁ (shiftFunctor C 1) = hom₃ ≫ B.whiskerLeft π₃Toπ₁
x✝¹ x✝ : J
φ : x✝¹ ⟶ x✝
⊢ (A.map φ ≫
        { hom₁ := hom₁.app x✝, hom₂ := hom₂.app x✝, hom₃ := hom₃.app x✝, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }).hom₃ =
    ({ hom₁ := hom₁.app x✝¹, hom₂ := hom₂.app x✝¹, hom₃ := hom₃.app x✝¹, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ } ≫
        B.map φ).hom₃