Module: Mathlib.CategoryTheory.Adjunction.Mates

Goal state before:
pp: 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
⊢ (mateEquiv adj₁ adj₃) (α ≫ₕ β) = (mateEquiv adj₁ adj₂) α ≫ᵥ (mateEquiv adj₂ adj₃) β

Tactic:
unfold hComp vComp mateEquiv

Goal state after:
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
⊢ {
        toFun := fun α ↦
          TwoSquare.mk R₁ (H₁ ⋙ H₂) (G₁ ⋙ G₂) R₃
            ((R₁ ⋙ G₁ ⋙ G₂).rightUnitor.inv ≫
              (R₁ ⋙ G₁ ⋙ G₂).whiskerLeft adj₃.unit ≫
                (R₁.associator (G₁ ⋙ G₂) (L₃ ⋙ R₃)).hom ≫
                  R₁.whiskerLeft ((G₁ ⋙ G₂).associator L₃ R₃).inv ≫
                    R₁.whiskerLeft (whiskerRight α.natTrans R₃) ≫
                      R₁.whiskerLeft (L₁.associator (H₁ ⋙ H₂) R₃).hom ≫
                        (R₁.associator L₁ ((H₁ ⋙ H₂) ⋙ R₃)).inv ≫
                          whiskerRight adj₁.counit ((H₁ ⋙ H₂) ⋙ R₃) ≫ ((H₁ ⋙ H₂) ⋙ R₃).leftUnitor.hom),
        invFun := fun β ↦
          TwoSquare.mk (G₁ ⋙ G₂) L₁ L₃ (H₁ ⋙ H₂)
            (((G₁ ⋙ G₂) ⋙ L₃).leftUnitor.inv ≫
              whiskerRight adj₁.unit ((G₁ ⋙ G₂) ⋙ L₃) ≫
                ((L₁ ⋙ R₁).associator (G₁ ⋙ G₂) L₃).inv ≫
                  whiskerRight (L₁.associator R₁ (G₁ ⋙ G₂)).hom L₃ ≫
                    whiskerRight (L₁.whiskerLeft β.natTrans) L₃ ≫
                      whiskerRight (L₁.associator (H₁ ⋙ H₂) R₃).inv L₃ ≫
                        ((L₁ ⋙ H₁ ⋙ H₂).associator R₃ L₃).hom ≫
                          (L₁ ⋙ H₁ ⋙ H₂).whiskerLeft adj₃.counit ≫ (L₁ ⋙ H₁ ⋙ H₂).rightUnitor.hom),
        left_inv := ⋯, right_inv := ⋯ }
      (TwoSquare.mk (G₁ ⋙ G₂) L₁ L₃ (H₁ ⋙ H₂)
        ((G₁.associator G₂ L₃).hom ≫
          G₁.whiskerLeft β.natTrans ≫
            (G₁.associator L₂ H₂).inv ≫ whiskerRight α.natTrans H₂ ≫ (L₁.associator H₁ H₂).hom)) =
    TwoSquare.mk R₁ (H₁ ⋙ H₂) (G₁ ⋙ G₂) R₃
      ((R₁.associator G₁ G₂).inv ≫
        whiskerRight
            ({
                  toFun := fun α ↦
                    TwoSquare.mk R₁ H₁ G₁ R₂
                      ((R₁ ⋙ G₁).rightUnitor.inv ≫
                        (R₁ ⋙ G₁).whiskerLeft adj₂.unit ≫
                          (R₁.associator G₁ (L₂ ⋙ R₂)).hom ≫
                            R₁.whiskerLeft (G₁.associator L₂ R₂).inv ≫
                              R₁.whiskerLeft (whiskerRight α.natTrans R₂) ≫
                                R₁.whiskerLeft (L₁.associator H₁ R₂).hom ≫
                                  (R₁.associator L₁ (H₁ ⋙ R₂)).inv ≫
                                    whiskerRight adj₁.counit (H₁ ⋙ R₂) ≫ (H₁ ⋙ R₂).leftUnitor.hom),
                  invFun := fun β ↦
                    TwoSquare.mk G₁ L₁ L₂ H₁
                      ((G₁ ⋙ L₂).leftUnitor.inv ≫
                        whiskerRight adj₁.unit (G₁ ⋙ L₂) ≫
                          ((L₁ ⋙ R₁).associator G₁ L₂).inv ≫
                            whiskerRight (L₁.associator R₁ G₁).hom L₂ ≫
                              whiskerRight (L₁.whiskerLeft β.natTrans) L₂ ≫
                                whiskerRight (L₁.associator H₁ R₂).inv L₂ ≫
                                  ((L₁ ⋙ H₁).associator R₂ L₂).hom ≫
                                    (L₁ ⋙ H₁).whiskerLeft adj₂.counit ≫ (L₁ ⋙ H₁).rightUnitor.hom),
                  left_inv := ⋯, right_inv := ⋯ }
                α).natTrans
            G₂ ≫
          (H₁.associator R₂ G₂).hom ≫
            H₁.whiskerLeft
                ({
                      toFun := fun α ↦
                        TwoSquare.mk R₂ H₂ G₂ R₃
                          ((R₂ ⋙ G₂).rightUnitor.inv ≫
                            (R₂ ⋙ G₂).whiskerLeft adj₃.unit ≫
                              (R₂.associator G₂ (L₃ ⋙ R₃)).hom ≫
                                R₂.whiskerLeft (G₂.associator L₃ R₃).inv ≫
                                  R₂.whiskerLeft (whiskerRight α.natTrans R₃) ≫
                                    R₂.whiskerLeft (L₂.associator H₂ R₃).hom ≫
                                      (R₂.associator L₂ (H₂ ⋙ R₃)).inv ≫
                                        whiskerRight adj₂.counit (H₂ ⋙ R₃) ≫ (H₂ ⋙ R₃).leftUnitor.hom),
                      invFun := fun β ↦
                        TwoSquare.mk G₂ L₂ L₃ H₂
                          ((G₂ ⋙ L₃).leftUnitor.inv ≫
                            whiskerRight adj₂.unit (G₂ ⋙ L₃) ≫
                              ((L₂ ⋙ R₂).associator G₂ L₃).inv ≫
                                whiskerRight (L₂.associator R₂ G₂).hom L₃ ≫
                                  whiskerRight (L₂.whiskerLeft β.natTrans) L₃ ≫
                                    whiskerRight (L₂.associator H₂ R₃).inv L₃ ≫
                                      ((L₂ ⋙ H₂).associator R₃ L₃).hom ≫
                                        (L₂ ⋙ H₂).whiskerLeft adj₃.counit ≫ (L₂ ⋙ H₂).rightUnitor.hom),
                      left_inv := ⋯, right_inv := ⋯ }
                    β).natTrans ≫
              (H₁.associator H₂ R₃).inv)