Module: Mathlib.CategoryTheory.Triangulated.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this✝ : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  F.map g ≫ (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
h₁' : f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂
h₂' : g ≫ (Functor.commShiftIso G 1).inv.app T.obj₁ = (adj.homEquiv Z T.obj₃) h ≫ G.map T.mor₃
this : Mono ((adj.homEquiv Z T.obj₃) h)
⊢ Triangle.mk (G.map T.mor₁) (G.map T.mor₂) (G.map T.mor₃ ≫ (Functor.commShiftIso G 1).hom.app T.obj₁) ∈
    distinguishedTriangles

Tactic:
have := isIso_of_yoneda_map_bijective (adj.homEquiv _ _ h) (fun Y => by
      constructor
      · intro φ₁ φ₂ hφ
        rw [← cancel_mono (adj.homEquiv _ _ h)]
        exact hφ
      · intro φ
        obtain ⟨ψ, hψ⟩ := Triangle.coyoneda_exact₁ _ mem (φ ≫ G.map T.mor₃ ≫
          (G.commShiftIso (1 : ℤ)).hom.app T.obj₁) (by
            dsimp
            rw [assoc, assoc, ← G.commShiftIso_hom_naturality, ← G.map_comp_assoc,
              comp_distTriang_mor_zero₃₁ _ hT, G.map_zero, zero_comp, comp_zero])
        dsimp at ψ hψ
        obtain ⟨α, hα⟩ : ∃ α, α = φ - ψ ≫ adj.homEquiv _ _ h := ⟨_, rfl⟩
        have hα₀ : α ≫ G.map T.mor₃ = 0 := by
          rw [hα, sub_comp, ← cancel_mono ((Functor.commShiftIso G (1 : ℤ)).hom.app T.obj₁),
            assoc, sub_comp, assoc, assoc, hψ, zero_comp, sub_eq_zero,
            ← cancel_mono ((Functor.commShiftIso G (1 : ℤ)).inv.app T.obj₁), assoc,
            assoc, assoc, assoc, h₂', Iso.hom_inv_id_app, comp_id]
        suffices ∃ (β : Y ⟶ Z), β ≫ adj.homEquiv _ _ h = α by
          obtain ⟨β, hβ⟩ := this
          refine ⟨ψ + β, ?_⟩
          dsimp
          rw [add_comp, hβ, hα, add_sub_cancel]
        obtain ⟨β, hβ⟩ := T.coyoneda_exact₃ hT ((adj.homEquiv _ _).symm α)
          ((adj.homEquiv _ _).injective (by simpa [homEquiv_unit, homEquiv_counit] using hα₀))
        refine ⟨adj.homEquiv _ _ β ≫ f, ?_⟩
        simpa [homEquiv_unit, h₁'] using congr_arg (adj.homEquiv _ _).toFun hβ.symm)

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasZeroObject C
inst✝¹² : HasZeroObject D
inst✝¹¹ : Preadditive C
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝³ : F.CommShift ℤ
inst✝² : G.CommShift ℤ
inst✝¹ : adj.CommShift ℤ
inst✝ : F.IsTriangulated
T : Triangle D
hT : T ∈ distinguishedTriangles
this✝¹ : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ (shiftFunctor C 1).obj (G.obj T.obj₁)
mem : Triangle.mk (G.map T.mor₁) f g ∈ distinguishedTriangles
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  F.map g ≫ (Functor.commShiftIso F 1).hom.app (G.obj T.obj₁) ≫ (shiftFunctor D 1).map (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
h₁' : f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂
h₂' : g ≫ (Functor.commShiftIso G 1).inv.app T.obj₁ = (adj.homEquiv Z T.obj₃) h ≫ G.map T.mor₃
this✝ : Mono ((adj.homEquiv Z T.obj₃) h)
this : IsIso ((adj.homEquiv Z T.obj₃) h)
⊢ Triangle.mk (G.map T.mor₁) (G.map T.mor₂) (G.map T.mor₃ ≫ (Functor.commShiftIso G 1).hom.app T.obj₁) ∈
    distinguishedTriangles