/-- A product of distinguished triangles is distinguished -/
lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
    (hT : ∀ j, T j ∈ distTriang C)
    [HasProduct (fun j => (T j).obj₁)] [HasProduct (fun j => (T j).obj₂)]
    [HasProduct (fun j => (T j).obj₃)] [HasProduct (fun j => (T j).obj₁⟦(1 : ℤ)⟧)] :
    productTriangle T ∈ distTriang C := by
  /- The proof proceeds by constructing a morphism of triangles
    `φ' : T' ⟶ productTriangle T` with `T'` distinguished, and such that
    `φ'.hom₁` and `φ'.hom₂` are identities. Then, it suffices to show that
    `φ'.hom₃` is an isomorphism, which is achieved by using Yoneda's lemma
    and diagram chases. -/
  let f₁ := Pi.map (fun j => (T j).mor₁)
  obtain ⟨Z, f₂, f₃, hT'⟩ := distinguished_cocone_triangle f₁
  let T' := Triangle.mk f₁ f₂ f₃
  change T' ∈ distTriang C at hT'
  let φ : ∀ j, T' ⟶ T j := fun j => completeDistinguishedTriangleMorphism _ _
    hT' (hT j) (Pi.π _ j) (Pi.π _ j) (by simp)
  let φ' := productTriangle.lift _ φ
  have h₁ : φ'.hom₁ = 𝟙 _ := by aesop_cat
  have h₂ : φ'.hom₂ = 𝟙 _ := by aesop_cat
  have : IsIso φ'.hom₁ := by rw [h₁]; infer_instance
  have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance
  suffices IsIso φ'.hom₃ by
    have : IsIso φ' := by
      apply Triangle.isIso_of_isIsos
      all_goals infer_instance
    exact isomorphic_distinguished _ hT' _ (asIso φ').symm
  refine' isIso_of_yoneda_map_bijective _ (fun A => ⟨_, _⟩)
  /- the proofs by diagram chase start here -/
  · suffices Mono φ'.hom₃ by
      intro a₁ a₂ ha
      simpa only [← cancel_mono φ'.hom₃] using ha
    rw [mono_iff_cancel_zero]
    intro A f hf
    have hf' : f ≫ T'.mor₃ = 0 := by
      rw [← cancel_mono (φ'.hom₁⟦1⟧'), zero_comp, assoc, φ'.comm₃, reassoc_of% hf, zero_comp]
    obtain ⟨g, hg⟩ := T'.coyoneda_exact₃ hT' f hf'
    have hg' : ∀ j, (g ≫ Pi.π _ j) ≫ (T j).mor₂ = 0 := fun j => by
      have : g ≫ T'.mor₂ ≫ φ'.hom₃ ≫ Pi.π _ j = 0 :=
        by rw [← reassoc_of% hg, reassoc_of% hf, zero_comp]
      rw [φ'.comm₂_assoc, h₂, id_comp] at this
      simpa using this
    have hg'' := fun j => (T j).coyoneda_exact₂ (hT j) _ (hg' j)
    let α := fun j => (hg'' j).choose
    have hα : ∀ j, _ = α j ≫ _ := fun j => (hg'' j).choose_spec
    

Proof state:
case intro.intro.intro.refine'_1.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ fun j => (T j).obj₁) ⟶ ∏ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' (_ : T j ∈ distinguishedTriangles) (Pi.π (fun j => (T j).obj₁) j)
      (Pi.π (fun j => (T j).obj₂) j)
      (_ :
        limMap (Discrete.natTrans fun X => (fun j => (T j).mor₁) X.as) ≫
            limit.π (Discrete.functor fun j => (T j).obj₂) { as := j } =
          Pi.π (fun j => (T j).obj₁) j ≫ (T j).mor₁)
φ' : T' ⟶ productTriangle fun j => T j := productTriangle.lift (fun j => T j) φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A✝ A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
hf' : f ≫ T'.mor₃ = 0
g : A ⟶ T'.obj₂
hg : f = g ≫ T'.mor₂
hg' : ∀ (j : J), (g ≫ Pi.π (fun j => (T j).obj₂) j) ≫ (T j).mor₂ = 0
hg'' : ∀ (j : J), ∃ g_1, g ≫ Pi.π (fun j => (T j).obj₂) j = g_1 ≫ (T j).mor₁
α : (j : J) → A ⟶ (T j).obj₁ := fun j => Exists.choose (_ : ∃ g_1, g ≫ Pi.π (fun j => (T j).obj₂) j = g_1 ≫ (T j).mor₁)
hα : ∀ (j : J), g ≫ Pi.π (fun j => (T j).obj₂) j = α j ≫ (T j).mor₁
⊢ f = 0

Next tactic:
have hg''' : g = Pi.lift α ≫ T'.mor₁ := by dsimp; ext j; rw [hα]; simp