-- Module: Mathlib.CategoryTheory.Triangulated.Adjunction
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : HasZeroObject.{v_1, u_1} C
inst_3 : HasZeroObject.{v_2, u_2} D
inst_4 : Preadditive.{v_1, u_1} C
inst_5 : Preadditive.{v_2, u_2} D
inst_6 : HasShift.{v_1, u_1, 0} C ℤ
inst_7 : HasShift.{v_2, u_2, 0} D ℤ
inst_8 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_9 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_10 : Pretriangulated.{v_1, u_1} C
inst_11 : Pretriangulated.{v_2, u_2} D
F : Functor.{v_1, v_2, u_1, u_2} C D
G : Functor.{v_2, v_1, u_2, u_1} D C
adj : F ⊣ G
inst_12 : F.CommShift ℤ
inst_13 : G.CommShift ℤ
inst_14 : adj.CommShift ℤ
inst_15 : F.IsTriangulated
T : Triangle.{v_2, u_2} D
hT : Membership.mem (γ := Set (Triangle.{v_2, u_2} D)) (@distinguishedTriangles _ _ inst_3 _ inst_5 inst_9 _) T
this_1 : G.Additive
Z : C
f : G.obj T.obj₂ ⟶ Z
g : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₁)
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_4 inst_8 _)
    (Triangle.mk (G.map T.mor₁) f g)
h : F.obj Z ⟶ T.obj₃
h₁ : F.map f ≫ h = adj.counit.app T.obj₂ ≫ T.mor₂
h₂ :
  F.map g ≫
      NatTrans.app (Iso.hom (Functor.commShiftIso F 1)) (G.obj T.obj₁) ≫
        Functor.map (shiftFunctor D (A := ℤ) 1) (adj.counit.app T.obj₁) =
    h ≫ T.mor₃
h₁' : f ≫ adj.unit.app Z ≫ G.map h = G.map T.mor₂
h₂' :
  Eq (α := Z ⟶ (shiftFunctor D (A := ℤ) 1 ⋙ G).obj T.obj₁)
    (g ≫ NatTrans.app (Iso.inv (Functor.commShiftIso G 1)) T.obj₁)
    (((adj.homEquiv Z T.obj₃ : Equiv.{v_2 + 1, v_1 + 1} (F.obj Z ⟶ T.obj₃) (Z ⟶ G.obj T.obj₃)) :
          (a : F.obj Z ⟶ T.obj₃) → Z ⟶ G.obj T.obj₃)
        h ≫
      G.map T.mor₃)
this :
  Mono.{v_1, u_1} (X := Z)
    (((adj.homEquiv Z T.obj₃ : Equiv.{v_2 + 1, v_1 + 1} (F.obj Z ⟶ T.obj₃) (Z ⟶ G.obj T.obj₃)) :
          (a : F.obj Z ⟶ T.obj₃) → Z ⟶ G.obj T.obj₃)
        h :
      Z ⟶ G.obj T.obj₃)
Y : C
φ : Y ⟶ G.obj T.obj₃
rw :
  Eq (α := Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₂))
    (φ ≫
      (G.map T.mor₃ ≫ NatTrans.app (Iso.hom (Functor.commShiftIso G 1)) T.obj₁) ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) (G.map T.mor₁))
    0
rw₁ :
  Eq (α := Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₂))
    (φ ≫
      G.map T.mor₃ ≫
        NatTrans.app (Iso.hom (Functor.commShiftIso G 1)) T.obj₁ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (G.map T.mor₁))
    0
rw₂ :
  Eq (α := Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₂))
    (φ ≫
      G.map T.mor₃ ≫
        G.map (Functor.map (shiftFunctor D (A := ℤ) 1) T.mor₁) ≫
          NatTrans.app (Iso.hom (Functor.commShiftIso G 1)) T.obj₂)
    0
rw₃ :
  Eq (α := Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₂))
    (φ ≫
      G.map (T.mor₃ ≫ Functor.map (shiftFunctor D (A := ℤ) 1) T.mor₁) ≫
        NatTrans.app (Iso.hom (Functor.commShiftIso G 1)) T.obj₂)
    0
rw₄ :
  Eq (α := Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₂))
    (φ ≫
      CategoryStruct.comp (Y := G.obj (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D (A := ℤ) 1) T.obj₂)) (G.map 0)
        (NatTrans.app (Iso.hom (Functor.commShiftIso G 1)) T.obj₂))
    0
rw₅ :
  Eq (α := Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₂))
    (φ ≫
      CategoryStruct.comp (Y := G.obj (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D (A := ℤ) 1) T.obj₂)) 0
        (NatTrans.app (Iso.hom (Functor.commShiftIso G 1)) T.obj₂))
    0
rw₆ : Eq (α := Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₂)) (φ ≫ 0) 0
rw₇ rw₈ : Eq.{v_1 + 1} (α := Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₂)) 0 0
⊢ Eq (α := Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (G.obj T.obj₂))
    ((φ ≫ G.map T.mor₃ ≫ NatTrans.app (Iso.hom (Functor.commShiftIso G 1)) T.obj₁) ≫
      Functor.map (shiftFunctor C (A := ℤ) 1) (G.map T.mor₁))
    0

Tactic:
rw [assoc, assoc, ← G.commShiftIso_hom_naturality, ← G.map_comp_assoc,
              comp_distTriang_mor_zero₃₁ _ hT, G.map_zero, zero_comp, comp_zero]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.comp_zero, CategoryTheory.Functor.map_zero, CategoryTheory.Pretriangulated.comp_distTriang_mor_zero₃₁, CategoryTheory.Functor.commShiftIso_hom_naturality, CategoryTheory.Functor.map_comp_assoc, CategoryTheory.Limits.zero_comp