-- Mathlib/CategoryTheory/Adjunction/Mates.lean
theorem mateEquiv_vcomp (α : TwoSquare G₁ L₁ L₂ H₁) (β : TwoSquare G₂ L₂ L₃ H₂) :
    (mateEquiv adj₁ adj₃) (α ≫ₕ β) = (mateEquiv adj₁ adj₂ α) ≫ᵥ (mateEquiv adj₂ adj₃ β)

Proof state:
case h
A : Type u₁
B : Type u₂
C : Type u₃
D : Type u₄
E : Type u₅
F : Type u₆
inst✝⁵ : Category.{v₁, u₁} A
inst✝⁴ : Category.{v₂, u₂} B
inst✝³ : Category.{v₃, u₃} C
inst✝² : Category.{v₄, u₄} D
inst✝¹ : Category.{v₅, u₅} E
inst✝ : Category.{v₆, u₆} F
G₁ : A ⥤ C
G₂ : C ⥤ E
H₁ : B ⥤ D
H₂ : D ⥤ F
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : C ⥤ D
R₂ : D ⥤ C
L₃ : E ⥤ F
R₃ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : TwoSquare G₁ L₁ L₂ H₁
β : TwoSquare G₂ L₂ L₃ H₂
b : B
⊢ ({
              toFun := fun α =>
                TwoSquare.mk R₁ (H₁ ⋙ H₂) (G₁ ⋙ G₂) R₃
                  (whiskerLeft (R₁ ⋙ G₁ ⋙ G₂) adj₃.unit ≫
                    whiskerRight (whiskerLeft R₁ α.natTrans) R₃ ≫ whiskerRight adj₁.counit ((H₁ ⋙ H₂) ⋙ R₃)),
              invFun := fun β =>
                TwoSquare.mk (G₁ ⋙ G₂) L₁ L₃ (H₁ ⋙ H₂)
                  (whiskerRight adj₁.unit ((G₁ ⋙ G₂) ⋙ L₃) ≫
                    whiskerRight (whiskerLeft L₁ β.natTrans) L₃ ≫ whiskerLeft (L₁ ⋙ H₁ ⋙ H₂) adj₃.counit),
              left_inv := ⋯, right_inv := ⋯ }
            (TwoSquare.mk (G₁ ⋙ G₂) L₁ L₃ (H₁ ⋙ H₂)
              ((G₁.associator G₂ L₃).hom ≫
                whiskerLeft G₁ β.natTrans ≫
                  (G₁.associator L₂ H₂).inv ≫ whiskerRight α.natTrans H₂ ≫ (L₁.associator H₁ H₂).hom))).natTrans.app
      b =
    (TwoSquare.mk R₁ (H₁ ⋙ H₂) (G₁ ⋙ G₂) R₃
            ((R₁.associator G₁ G₂).hom ≫
              whiskerRight
                  ({
                        toFun := fun α =>
                          TwoSquare.mk R₁ H₁ G₁ R₂
                            (whiskerLeft (R₁ ⋙ G₁) adj₂.unit ≫
                              whiskerRight (whiskerLeft R₁ α.natTrans) R₂ ≫ whiskerRight adj₁.counit (H₁ ⋙ R₂)),
                        invFun := fun β =>
                          TwoSquare.mk G₁ L₁ L₂ H₁
                            (whiskerRight adj₁.unit (G₁ ⋙ L₂) ≫
                              whiskerRight (whiskerLeft L₁ β.natTrans) L₂ ≫ whiskerLeft (L₁ ⋙ H₁) adj₂.counit),
                        left_inv := ⋯, right_inv := ⋯ }
                      α).natTrans
                  G₂ ≫
                (H₁.associator R₂ G₂).inv ≫
                  whiskerLeft H₁
                      ({
                            toFun := fun α =>
                              TwoSquare.mk R₂ H₂ G₂ R₃
                                (whiskerLeft (R₂ ⋙ G₂) adj₃.unit ≫
                                  whiskerRight (whiskerLeft R₂ α.natTrans) R₃ ≫ whiskerRight adj₂.counit (H₂ ⋙ R₃)),
                            invFun := fun β =>
                              TwoSquare.mk G₂ L₂ L₃ H₂
                                (whiskerRight adj₂.unit (G₂ ⋙ L₃) ≫
                                  whiskerRight (whiskerLeft L₂ β.natTrans) L₃ ≫ whiskerLeft (L₂ ⋙ H₂) adj₃.counit),
                            left_inv := ⋯, right_inv := ⋯ }
                          β).natTrans ≫
                    (H₁.associator H₂ R₃).hom)).natTrans.app
      b

Tactic:
simp only [comp_obj, Equiv.coe_fn_mk, whiskerLeft_comp, whiskerLeft_twice, whiskerRight_comp,
  assoc, comp_app, whiskerLeft_app, whiskerRight_app, associator_hom_app, map_id,
  associator_inv_app, id_obj, Functor.comp_map, id_comp, whiskerRight_twice, comp_id]

Result:
case h
A : Type u₁
B : Type u₂
C : Type u₃
D : Type u₄
E : Type u₅
F : Type u₆
inst✝⁵ : Category.{v₁, u₁} A
inst✝⁴ : Category.{v₂, u₂} B
inst✝³ : Category.{v₃, u₃} C
inst✝² : Category.{v₄, u₄} D
inst✝¹ : Category.{v₅, u₅} E
inst✝ : Category.{v₆, u₆} F
G₁ : A ⥤ C
G₂ : C ⥤ E
H₁ : B ⥤ D
H₂ : D ⥤ F
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : C ⥤ D
R₂ : D ⥤ C
L₃ : E ⥤ F
R₃ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : TwoSquare G₁ L₁ L₂ H₁
β : TwoSquare G₂ L₂ L₃ H₂
b : B
⊢ adj₃.unit.app (G₂.obj (G₁.obj (R₁.obj b))) ≫
      R₃.map (β.natTrans.app (G₁.obj (R₁.obj b))) ≫
        R₃.map (H₂.map (α.natTrans.app (R₁.obj b))) ≫ R₃.map (H₂.map (H₁.map (adj₁.counit.app b))) =
    G₂.map (adj₂.unit.app (G₁.obj (R₁.obj b))) ≫
      G₂.map (R₂.map (α.natTrans.app (R₁.obj b))) ≫
        G₂.map (R₂.map (H₁.map (adj₁.counit.app b))) ≫
          adj₃.unit.app (G₂.obj (R₂.obj (H₁.obj b))) ≫
            R₃.map (β.natTrans.app (R₂.obj (H₁.obj b))) ≫ R₃.map (H₂.map (adj₂.counit.app (H₁.obj b)))

Proof state:
case h
A : Type u₁
B : Type u₂
C : Type u₃
D : Type u₄
E : Type u₅
F : Type u₆
inst✝⁵ : Category.{v₁, u₁} A
inst✝⁴ : Category.{v₂, u₂} B
inst✝³ : Category.{v₃, u₃} C
inst✝² : Category.{v₄, u₄} D
inst✝¹ : Category.{v₅, u₅} E
inst✝ : Category.{v₆, u₆} F
G₁ : A ⥤ C
G₂ : C ⥤ E
H₁ : B ⥤ D
H₂ : D ⥤ F
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : C ⥤ D
R₂ : D ⥤ C
L₃ : E ⥤ F
R₃ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : TwoSquare G₁ L₁ L₂ H₁
β : TwoSquare G₂ L₂ L₃ H₂
b : B
⊢ adj₃.unit.app (G₂.obj (G₁.obj (R₁.obj b))) ≫
      R₃.map (β.natTrans.app (G₁.obj (R₁.obj b))) ≫
        R₃.map (H₂.map (α.natTrans.app (R₁.obj b))) ≫ R₃.map (H₂.map (H₁.map (adj₁.counit.app b))) =
    adj₃.unit.app (G₂.obj (G₁.obj (R₁.obj b))) ≫
      R₃.map (β.app (G₁.obj (R₁.obj b))) ≫
        R₃.map ((L₂ ⋙ H₂).map (adj₂.unit.app (G₁.obj (R₁.obj b)) ≫ R₂.map (α.natTrans.app (R₁.obj b)))) ≫
          R₃.map (H₂.map (adj₂.counit.app (H₁.obj (L₁.obj (R₁.obj b))) ≫ (𝟭 D).map (H₁.map (adj₁.counit.app b))))

Tactic:
simp only [comp_obj, Functor.comp_map, map_comp, id_obj, Functor.id_map, assoc]

Result:
case h
A : Type u₁
B : Type u₂
C : Type u₃
D : Type u₄
E : Type u₅
F : Type u₆
inst✝⁵ : Category.{v₁, u₁} A
inst✝⁴ : Category.{v₂, u₂} B
inst✝³ : Category.{v₃, u₃} C
inst✝² : Category.{v₄, u₄} D
inst✝¹ : Category.{v₅, u₅} E
inst✝ : Category.{v₆, u₆} F
G₁ : A ⥤ C
G₂ : C ⥤ E
H₁ : B ⥤ D
H₂ : D ⥤ F
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : C ⥤ D
R₂ : D ⥤ C
L₃ : E ⥤ F
R₃ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : TwoSquare G₁ L₁ L₂ H₁
β : TwoSquare G₂ L₂ L₃ H₂
b : B
⊢ adj₃.unit.app (G₂.obj (G₁.obj (R₁.obj b))) ≫
      R₃.map (β.natTrans.app (G₁.obj (R₁.obj b))) ≫
        R₃.map (H₂.map (α.natTrans.app (R₁.obj b))) ≫ R₃.map (H₂.map (H₁.map (adj₁.counit.app b))) =
    adj₃.unit.app (G₂.obj (G₁.obj (R₁.obj b))) ≫
      R₃.map (β.app (G₁.obj (R₁.obj b))) ≫
        R₃.map (H₂.map (L₂.map (adj₂.unit.app (G₁.obj (R₁.obj b))))) ≫
          R₃.map (H₂.map (L₂.map (R₂.map (α.natTrans.app (R₁.obj b))))) ≫
            R₃.map (H₂.map (adj₂.counit.app (H₁.obj (L₁.obj (R₁.obj b))))) ≫
              R₃.map (H₂.map (H₁.map (adj₁.counit.app b)))