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
G : A ⥤ D
H : B ⥤ E
K : C ⥤ F
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : D ⥤ E
R₂ : E ⥤ D
L₃ : B ⥤ C
R₃ : C ⥤ B
L₄ : E ⥤ F
R₄ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
adj₄ : L₄ ⊣ R₄
α : TwoSquare G L₁ L₂ H
β : TwoSquare H L₃ L₄ K
⊢ (mateEquiv (adj₁.comp adj₃) (adj₂.comp adj₄)) (α ≫ᵥ β) = (mateEquiv adj₃ adj₄) β ≫ₕ (mateEquiv adj₁ adj₂) α

Tactic:
unfold vComp hComp mateEquiv Adjunction.comp

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
G : A ⥤ D
H : B ⥤ E
K : C ⥤ F
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : D ⥤ E
R₂ : E ⥤ D
L₃ : B ⥤ C
R₃ : C ⥤ B
L₄ : E ⥤ F
R₄ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
adj₄ : L₄ ⊣ R₄
α : TwoSquare G L₁ L₂ H
β : TwoSquare H L₃ L₄ K
⊢ {
        toFun := fun α ↦
          TwoSquare.mk (R₃ ⋙ R₁) K G (R₄ ⋙ R₂)
            (((R₃ ⋙ R₁) ⋙ G).rightUnitor.inv ≫
              ((R₃ ⋙ R₁) ⋙ G).whiskerLeft
                  (Adjunction.mk'
                      { homEquiv := fun x x_1 ↦ (adj₄.homEquiv (L₂.obj x) x_1).trans (adj₂.homEquiv x (R₄.obj x_1)),
                        unit :=
                          adj₂.unit ≫
                            whiskerRight (L₂.rightUnitor.inv ≫ L₂.whiskerLeft adj₄.unit ≫ (L₂.associator L₄ R₄).inv)
                                R₂ ≫
                              ((L₂ ⋙ L₄).associator R₄ R₂).hom,
                        counit :=
                          ((R₄ ⋙ R₂).associator L₂ L₄).inv ≫
                            whiskerRight ((R₄.associator R₂ L₂).hom ≫ R₄.whiskerLeft adj₂.counit ≫ R₄.rightUnitor.hom)
                                L₄ ≫
                              adj₄.counit,
                        homEquiv_unit := ⋯, homEquiv_counit := ⋯ }).unit ≫
                ((R₃ ⋙ R₁).associator G ((L₂ ⋙ L₄) ⋙ R₄ ⋙ R₂)).hom ≫
                  (R₃ ⋙ R₁).whiskerLeft (G.associator (L₂ ⋙ L₄) (R₄ ⋙ R₂)).inv ≫
                    (R₃ ⋙ R₁).whiskerLeft (whiskerRight α.natTrans (R₄ ⋙ R₂)) ≫
                      (R₃ ⋙ R₁).whiskerLeft ((L₁ ⋙ L₃).associator K (R₄ ⋙ R₂)).hom ≫
                        ((R₃ ⋙ R₁).associator (L₁ ⋙ L₃) (K ⋙ R₄ ⋙ R₂)).inv ≫
                          whiskerRight
                              (Adjunction.mk'
                                  {
                                    homEquiv := fun x x_1 ↦
                                      (adj₃.homEquiv (L₁.obj x) x_1).trans (adj₁.homEquiv x (R₃.obj x_1)),
                                    unit :=
                                      adj₁.unit ≫
                                        whiskerRight
                                            (L₁.rightUnitor.inv ≫ L₁.whiskerLeft adj₃.unit ≫ (L₁.associator L₃ R₃).inv)
                                            R₁ ≫
                                          ((L₁ ⋙ L₃).associator R₃ R₁).hom,
                                    counit :=
                                      ((R₃ ⋙ R₁).associator L₁ L₃).inv ≫
                                        whiskerRight
                                            ((R₃.associator R₁ L₁).hom ≫
                                              R₃.whiskerLeft adj₁.counit ≫ R₃.rightUnitor.hom)
                                            L₃ ≫
                                          adj₃.counit,
                                    homEquiv_unit := ⋯, homEquiv_counit := ⋯ }).counit
                              (K ⋙ R₄ ⋙ R₂) ≫
                            (K ⋙ R₄ ⋙ R₂).leftUnitor.hom),
        invFun := fun β ↦
          TwoSquare.mk G (L₁ ⋙ L₃) (L₂ ⋙ L₄) K
            ((G ⋙ L₂ ⋙ L₄).leftUnitor.inv ≫
              whiskerRight
                  (Adjunction.mk'
                      { homEquiv := fun x x_1 ↦ (adj₃.homEquiv (L₁.obj x) x_1).trans (adj₁.homEquiv x (R₃.obj x_1)),
                        unit :=
                          adj₁.unit ≫
                            whiskerRight (L₁.rightUnitor.inv ≫ L₁.whiskerLeft adj₃.unit ≫ (L₁.associator L₃ R₃).inv)
                                R₁ ≫
                              ((L₁ ⋙ L₃).associator R₃ R₁).hom,
                        counit :=
                          ((R₃ ⋙ R₁).associator L₁ L₃).inv ≫
                            whiskerRight ((R₃.associator R₁ L₁).hom ≫ R₃.whiskerLeft adj₁.counit ≫ R₃.rightUnitor.hom)
                                L₃ ≫
                              adj₃.counit,
                        homEquiv_unit := ⋯, homEquiv_counit := ⋯ }).unit
                  (G ⋙ L₂ ⋙ L₄) ≫
                (((L₁ ⋙ L₃) ⋙ R₃ ⋙ R₁).associator G (L₂ ⋙ L₄)).inv ≫
                  whiskerRight ((L₁ ⋙ L₃).associator (R₃ ⋙ R₁) G).hom (L₂ ⋙ L₄) ≫
                    whiskerRight ((L₁ ⋙ L₃).whiskerLeft β.natTrans) (L₂ ⋙ L₄) ≫
                      whiskerRight ((L₁ ⋙ L₃).associator K (R₄ ⋙ R₂)).inv (L₂ ⋙ L₄) ≫
                        (((L₁ ⋙ L₃) ⋙ K).associator (R₄ ⋙ R₂) (L₂ ⋙ L₄)).hom ≫
                          ((L₁ ⋙ L₃) ⋙ K).whiskerLeft
                              (Adjunction.mk'
                                  {
                                    homEquiv := fun x x_1 ↦
                                      (adj₄.homEquiv (L₂.obj x) x_1).trans (adj₂.homEquiv x (R₄.obj x_1)),
                                    unit :=
                                      adj₂.unit ≫
                                        whiskerRight
                                            (L₂.rightUnitor.inv ≫ L₂.whiskerLeft adj₄.unit ≫ (L₂.associator L₄ R₄).inv)
                                            R₂ ≫
                                          ((L₂ ⋙ L₄).associator R₄ R₂).hom,
                                    counit :=
                                      ((R₄ ⋙ R₂).associator L₂ L₄).inv ≫
                                        whiskerRight
                                            ((R₄.associator R₂ L₂).hom ≫
                                              R₄.whiskerLeft adj₂.counit ≫ R₄.rightUnitor.hom)
                                            L₄ ≫
                                          adj₄.counit,
                                    homEquiv_unit := ⋯, homEquiv_counit := ⋯ }).counit ≫
                            ((L₁ ⋙ L₃) ⋙ K).rightUnitor.hom),
        left_inv := ⋯, right_inv := ⋯ }
      (TwoSquare.mk G (L₁ ⋙ L₃) (L₂ ⋙ L₄) K
        ((G.associator L₂ L₄).inv ≫
          whiskerRight α.natTrans L₄ ≫
            (L₁.associator H L₄).hom ≫ L₁.whiskerLeft β.natTrans ≫ (L₁.associator L₃ K).inv)) =
    TwoSquare.mk (R₃ ⋙ R₁) K G (R₄ ⋙ R₂)
      ((R₃.associator R₁ G).hom ≫
        R₃.whiskerLeft
            ({
                  toFun := fun α ↦
                    TwoSquare.mk R₁ H G R₂
                      ((R₁ ⋙ G).rightUnitor.inv ≫
                        (R₁ ⋙ G).whiskerLeft adj₂.unit ≫
                          (R₁.associator G (L₂ ⋙ R₂)).hom ≫
                            R₁.whiskerLeft (G.associator L₂ R₂).inv ≫
                              R₁.whiskerLeft (whiskerRight α.natTrans R₂) ≫
                                R₁.whiskerLeft (L₁.associator H R₂).hom ≫
                                  (R₁.associator L₁ (H ⋙ R₂)).inv ≫
                                    whiskerRight adj₁.counit (H ⋙ R₂) ≫ (H ⋙ R₂).leftUnitor.hom),
                  invFun := fun β ↦
                    TwoSquare.mk G L₁ L₂ H
                      ((G ⋙ L₂).leftUnitor.inv ≫
                        whiskerRight adj₁.unit (G ⋙ L₂) ≫
                          ((L₁ ⋙ R₁).associator G L₂).inv ≫
                            whiskerRight (L₁.associator R₁ G).hom L₂ ≫
                              whiskerRight (L₁.whiskerLeft β.natTrans) L₂ ≫
                                whiskerRight (L₁.associator H R₂).inv L₂ ≫
                                  ((L₁ ⋙ H).associator R₂ L₂).hom ≫
                                    (L₁ ⋙ H).whiskerLeft adj₂.counit ≫ (L₁ ⋙ H).rightUnitor.hom),
                  left_inv := ⋯, right_inv := ⋯ }
                α).natTrans ≫
          (R₃.associator H R₂).inv ≫
            whiskerRight
                ({
                      toFun := fun α ↦
                        TwoSquare.mk R₃ K H R₄
                          ((R₃ ⋙ H).rightUnitor.inv ≫
                            (R₃ ⋙ H).whiskerLeft adj₄.unit ≫
                              (R₃.associator H (L₄ ⋙ R₄)).hom ≫
                                R₃.whiskerLeft (H.associator L₄ R₄).inv ≫
                                  R₃.whiskerLeft (whiskerRight α.natTrans R₄) ≫
                                    R₃.whiskerLeft (L₃.associator K R₄).hom ≫
                                      (R₃.associator L₃ (K ⋙ R₄)).inv ≫
                                        whiskerRight adj₃.counit (K ⋙ R₄) ≫ (K ⋙ R₄).leftUnitor.hom),
                      invFun := fun β ↦
                        TwoSquare.mk H L₃ L₄ K
                          ((H ⋙ L₄).leftUnitor.inv ≫
                            whiskerRight adj₃.unit (H ⋙ L₄) ≫
                              ((L₃ ⋙ R₃).associator H L₄).inv ≫
                                whiskerRight (L₃.associator R₃ H).hom L₄ ≫
                                  whiskerRight (L₃.whiskerLeft β.natTrans) L₄ ≫
                                    whiskerRight (L₃.associator K R₄).inv L₄ ≫
                                      ((L₃ ⋙ K).associator R₄ L₄).hom ≫
                                        (L₃ ⋙ K).whiskerLeft adj₄.counit ≫ (L₃ ⋙ K).rightUnitor.hom),
                      left_inv := ⋯, right_inv := ⋯ }
                    β).natTrans
                R₂ ≫
              (K.associator R₄ R₂).hom)