Module: Mathlib.CategoryTheory.Adjunction.Mates

Goal state before:
pp: A : Type u₁
B : Type u₂
C : Type u₃
D : Type u₄
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} B
inst✝¹ : Category.{v₃, u₃} C
inst✝ : Category.{v₄, u₄} D
G : A ⥤ C
H : B ⥤ D
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : C ⥤ D
R₂ : D ⥤ C
L₃ : C ⥤ D
R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : TwoSquare G L₁ L₂ H
β : L₃ ⟶ L₂
b : B
vcomp :
  (mateEquiv adj₁ adj₃)
      (TwoSquare.mk (G ⋙ 𝟭 C) L₁ L₃ (H ⋙ 𝟭 D)
        ((G.associator (𝟭 C) L₃).hom ≫
          G.whiskerLeft (natTrans (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)) ≫
            (G.associator L₂ (𝟭 D)).inv ≫ whiskerRight α.natTrans (𝟭 D) ≫ (L₁.associator H (𝟭 D)).hom)) =
    TwoSquare.mk R₁ (H ⋙ 𝟭 D) (G ⋙ 𝟭 C) R₃
      ((R₁.associator G (𝟭 C)).inv ≫
        whiskerRight ((mateEquiv adj₁ adj₂) α).natTrans (𝟭 C) ≫
          (H.associator R₂ (𝟭 C)).hom ≫
            H.whiskerLeft ((mateEquiv adj₂ adj₃) (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)).natTrans ≫
              (H.associator (𝟭 D) R₃).inv)
vcompb :
  ((mateEquiv adj₁ adj₃)
          (TwoSquare.mk (G ⋙ 𝟭 C) L₁ L₃ (H ⋙ 𝟭 D)
            ((G.associator (𝟭 C) L₃).hom ≫
              G.whiskerLeft (natTrans (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)) ≫
                (G.associator L₂ (𝟭 D)).inv ≫ whiskerRight α.natTrans (𝟭 D) ≫ (L₁.associator H (𝟭 D)).hom))).app
      b =
    (TwoSquare.mk R₁ (H ⋙ 𝟭 D) (G ⋙ 𝟭 C) R₃
          ((R₁.associator G (𝟭 C)).inv ≫
            whiskerRight ((mateEquiv adj₁ adj₂) α).natTrans (𝟭 C) ≫
              (H.associator R₂ (𝟭 C)).hom ≫
                H.whiskerLeft ((mateEquiv adj₂ adj₃) (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)).natTrans ≫
                  (H.associator (𝟭 D) R₃).inv)).app
      b
⊢ ((mateEquiv adj₁ adj₃) (α.whiskerRight β)).natTrans.app b =
    (((mateEquiv adj₁ adj₂) α).whiskerBottom ((conjugateEquiv adj₂ adj₃) β)).natTrans.app b

Tactic:
simp only [comp_obj, id_obj, whiskerLeft_comp, assoc, mateEquiv_apply, whiskerLeft_twice,
    Iso.hom_inv_id_assoc, whiskerRight_comp, comp_app, Functor.whiskerLeft_app,
    Functor.whiskerRight_app, associator_hom_app, map_id, associator_inv_app, leftUnitor_hom_app,
    rightUnitor_inv_app, Functor.id_map, Functor.comp_map, id_comp, whiskerRight_twice,
    comp_id] at vcompb

Goal state after:
A : Type u₁
B : Type u₂
C : Type u₃
D : Type u₄
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} B
inst✝¹ : Category.{v₃, u₃} C
inst✝ : Category.{v₄, u₄} D
G : A ⥤ C
H : B ⥤ D
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : C ⥤ D
R₂ : D ⥤ C
L₃ : C ⥤ D
R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : TwoSquare G L₁ L₂ H
β : L₃ ⟶ L₂
b : B
vcomp :
  (mateEquiv adj₁ adj₃)
      (TwoSquare.mk (G ⋙ 𝟭 C) L₁ L₃ (H ⋙ 𝟭 D)
        ((G.associator (𝟭 C) L₃).hom ≫
          G.whiskerLeft (natTrans (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)) ≫
            (G.associator L₂ (𝟭 D)).inv ≫ whiskerRight α.natTrans (𝟭 D) ≫ (L₁.associator H (𝟭 D)).hom)) =
    TwoSquare.mk R₁ (H ⋙ 𝟭 D) (G ⋙ 𝟭 C) R₃
      ((R₁.associator G (𝟭 C)).inv ≫
        whiskerRight ((mateEquiv adj₁ adj₂) α).natTrans (𝟭 C) ≫
          (H.associator R₂ (𝟭 C)).hom ≫
            H.whiskerLeft ((mateEquiv adj₂ adj₃) (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)).natTrans ≫
              (H.associator (𝟭 D) R₃).inv)
vcompb :
  adj₃.unit.app (G.obj (R₁.obj b)) ≫
      R₃.map (β.app (G.obj (R₁.obj b))) ≫ R₃.map (α.natTrans.app (R₁.obj b)) ≫ R₃.map (H.map (adj₁.counit.app b)) =
    adj₂.unit.app (G.obj (R₁.obj b)) ≫
      R₂.map (α.natTrans.app (R₁.obj b)) ≫
        R₂.map (H.map (adj₁.counit.app b)) ≫
          adj₃.unit.app (R₂.obj (H.obj b)) ≫ R₃.map (β.app (R₂.obj (H.obj b))) ≫ R₃.map (adj₂.counit.app (H.obj b))
⊢ ((mateEquiv adj₁ adj₃) (α.whiskerRight β)).natTrans.app b =
    (((mateEquiv adj₁ adj₂) α).whiskerBottom ((conjugateEquiv adj₂ adj₃) β)).natTrans.app b