-- Module: Mathlib.CategoryTheory.Triangulated.Triangulated
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
h :
  ∀ ⦃X₁' X₂' X₃' : C⦄ (u₁₂' : X₁' ⟶ X₂') (u₂₃' : X₂' ⟶ X₃'),
    ∃ (X₁ : C) (X₂ : C) (X₃ : C) (Z₁₂ : C) (Z₂₃ : C) (Z₁₃ : C) (u₁₂ : X₁ ⟶ X₂) (u₂₃ : X₂ ⟶ X₃) (e₁ : X₁' ≅ X₁) (e₂ :
      X₂' ≅ X₂) (e₃ : X₃' ≅ X₃) (_ : u₁₂' ≫ e₂.hom = e₁.hom ≫ u₁₂) (_ : u₂₃' ≫ e₃.hom = e₂.hom ≫ u₂₃) (v₁₂ : X₂ ⟶ Z₁₂)
      (w₁₂ : Z₁₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₁) (h₁₂ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
        (Triangle.mk u₁₂ v₁₂ w₁₂))
      (v₂₃ : X₃ ⟶ Z₂₃) (w₂₃ : Z₂₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₂) (h₂₃ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
        (Triangle.mk u₂₃ v₂₃ w₂₃))
      (v₁₃ : X₃ ⟶ Z₁₃) (w₁₃ : Z₁₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₁) (h₁₃ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
        (Triangle.mk (u₁₂ ≫ u₂₃) v₁₃ w₁₃)),
      Nonempty.{v_1 + 1} (Octahedron ⋯ h₁₂ h₂₃ h₁₃)
X₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C
u₁₂' : X₁' ⟶ X₂'
u₂₃' : X₂' ⟶ X₃'
u₁₃' : X₁' ⟶ X₃'
comm' : u₁₂' ≫ u₂₃' = u₁₃'
v₁₂' : X₂' ⟶ Z₁₂'
w₁₂' : Z₁₂' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₁'
h₁₂' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk u₁₂' v₁₂' w₁₂')
v₂₃' : X₃' ⟶ Z₂₃'
w₂₃' : Z₂₃' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₂'
h₂₃' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk u₂₃' v₂₃' w₂₃')
v₁₃' : X₃' ⟶ Z₁₃'
w₁₃' : Z₁₃' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₁'
h₁₃' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk u₁₃' v₁₃' w₁₃')
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
e₁ : X₁' ≅ X₁
e₂ : X₂' ≅ X₂
e₃ : X₃' ≅ X₃
comm₁₂ : u₁₂' ≫ e₂.hom = e₁.hom ≫ u₁₂
comm₂₃ : u₂₃' ≫ e₃.hom = e₂.hom ≫ u₂₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₁
h₁₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk u₁₂ v₁₂ w₁₂)
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₂
h₂₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk u₂₃ v₂₃ w₂₃)
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₁
h₁₃ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_1 inst_4 _)
    (Triangle.mk (u₁₂ ≫ u₂₃) v₁₃ w₁₃)
H : Nonempty.{v_1 + 1} (Octahedron ⋯ h₁₂ h₂₃ h₁₃)
⊢ Nonempty.{v_1 + 1} (Octahedron comm' h₁₂' h₂₃' h₁₃')

Tactic:
exact ⟨Octahedron.ofIso u₁₂' u₂₃' u₁₃' comm' h₁₂' h₂₃' h₁₃'
      u₁₂ u₂₃ _ rfl e₁ e₂ e₃ comm₁₂ comm₂₃ v₁₂ w₁₂ h₁₂ v₂₃ w₂₃ h₂₃ v₁₃ w₁₃ h₁₃ H.some⟩