-- Lean 4 proof step from Mathlib.CategoryTheory.Adjunction.Mates (goal)
Goal:
A : Type u₁
B : Type u₂
C : Type u₃
D : Type u₄
E : Type u₅
F : Type u₆
inst : Category.{v₁, u₁} A
inst_1 : Category.{v₂, u₂} B
inst_2 : Category.{v₃, u₃} C
inst_3 : Category.{v₄, u₄} D
inst_4 : Category.{v₅, u₅} E
inst_5 : Category.{v₆, u₆} F
G : Functor.{v₁, v₄, u₁, u₄} A D
H : Functor.{v₂, v₅, u₂, u₅} B E
K : Functor.{v₃, v₆, u₃, u₆} C F
L₁ : Functor.{v₁, v₂, u₁, u₂} A B
R₁ : Functor.{v₂, v₁, u₂, u₁} B A
L₂ : Functor.{v₄, v₅, u₄, u₅} D E
R₂ : Functor.{v₅, v₄, u₅, u₄} E D
L₃ : Functor.{v₂, v₃, u₂, u₃} B C
R₃ : Functor.{v₃, v₂, u₃, u₂} C B
L₄ : Functor.{v₅, v₆, u₅, u₆} E F
R₄ : Functor.{v₆, v₅, u₆, u₅} 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
c : C
⊢ app
      (natTrans
        (({
              toFun := fun (α : TwoSquare G (L₁ ⋙ L₃) (L₂ ⋙ L₄) K) ↦
                TwoSquare.mk (R₃ ⋙ R₁) K G (R₄ ⋙ R₂)
                  (((R₃ ⋙ R₁) ⋙ G).rightUnitor.inv ≫
                    ((R₃ ⋙ R₁) ⋙ G).whiskerLeft
                        (Adjunction.mk'
                            {
                              homEquiv := fun (x : D) (x_1 : F) ↦
                                Equiv.trans (adj₄.homEquiv (L₂.obj x) x_1) (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 := ⋯ :
                              CoreHomEquivUnitCounit (L₂ ⋙ L₄) (R₄ ⋙ R₂) }).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 : A) (x_1 : C) ↦
                                            Equiv.trans (adj₃.homEquiv (L₁.obj x) x_1) (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 := ⋯ :
                                          CoreHomEquivUnitCounit (L₁ ⋙ L₃) (R₃ ⋙ R₁) }).counit
                                    (K ⋙ R₄ ⋙ R₂) ≫
                                  (K ⋙ R₄ ⋙ R₂).leftUnitor.hom),
              invFun := fun (β : TwoSquare (R₃ ⋙ R₁) K G (R₄ ⋙ R₂)) ↦
                TwoSquare.mk G (L₁ ⋙ L₃) (L₂ ⋙ L₄) K
                  ((G ⋙ L₂ ⋙ L₄).leftUnitor.inv ≫
                    whiskerRight
                        (Adjunction.mk'
                            {
                              homEquiv := fun (x : A) (x_1 : C) ↦
                                Equiv.trans (adj₃.homEquiv (L₁.obj x) x_1) (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 := ⋯ :
                              CoreHomEquivUnitCounit (L₁ ⋙ L₃) (R₃ ⋙ R₁) }).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 : D) (x_1 : F) ↦
                                            Equiv.trans (adj₄.homEquiv (L₂.obj x) x_1) (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 := ⋯ :
                                          CoreHomEquivUnitCounit (L₂ ⋙ L₄) (R₄ ⋙ R₂) }).counit ≫
                                  ((L₁ ⋙ L₃) ⋙ K).rightUnitor.hom),
              left_inv := ⋯, right_inv := ⋯ } :
            (a : TwoSquare G (L₁ ⋙ L₃) (L₂ ⋙ L₄) K) → TwoSquare (R₃ ⋙ R₁) K G (R₄ ⋙ R₂))
          (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))))
      c =
    (TwoSquare.mk (R₃ ⋙ R₁) K G (R₄ ⋙ R₂)
            ((R₃.associator R₁ G).hom ≫
              R₃.whiskerLeft
                  (natTrans
                    (({
                          toFun := fun (α : TwoSquare G L₁ L₂ H) ↦
                            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 R₁ H G R₂) ↦
                            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 := ⋯ : TwoSquare G L₁ L₂ H ≃ TwoSquare R₁ H G R₂ } :
                        (a : TwoSquare G L₁ L₂ H) → TwoSquare R₁ H G R₂)
                      α)) ≫
                (R₃.associator H R₂).inv ≫
                  whiskerRight
                      (natTrans
                        (({
                              toFun := fun (α : TwoSquare H L₃ L₄ K) ↦
                                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 R₃ K H R₄) ↦
                                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 := ⋯ : TwoSquare H L₃ L₄ K ≃ TwoSquare R₃ K H R₄ } :
                            (a : TwoSquare H L₃ L₄ K) → TwoSquare R₃ K H R₄)
                          β))
                      R₂ ≫
                    (K.associator R₄ R₂).hom)).natTrans.app
      c
have h : adj₂.unit.app (G.obj (R₁.obj (R₃.obj c))) ≫
    R₂.map (adj₄.unit.app (L₂.obj (G.obj (R₁.obj (R₃.obj c))))) ≫
      R₂.map (R₄.map (L₄.map (α.natTrans.app (R₁.obj (R₃.obj c))))) ≫
        R₂.map (R₄.map (β.natTrans.app (L₁.obj (R₁.obj (R₃.obj c))))) ≫
          R₂.map (R₄.map (K.map (L₃.map (adj₁.counit.app (R₃.obj c))))) ≫ R₂.map (R₄.map (K.map (adj₃.counit.app c))) =
  adj₂.unit.app (G.obj (R₁.obj (R₃.obj c))) ≫
    R₂.map (α.natTrans.app (R₁.obj (R₃.obj c))) ≫
      R₂.map (H.map (adj₁.counit.app (R₃.obj c))) ≫
        R₂.map (adj₄.unit.app (H.obj (R₃.obj c))) ≫
          R₂.map (R₄.map (β.natTrans.app (R₃.obj c))) ≫ R₂.map (R₄.map (K.map (adj₃.counit.app c)))
Tactic: simp only [comp_obj, whiskerRight_comp, assoc, mk'_unit, whiskerLeft_comp, mk'_counit,
    whiskerRight_twice, Iso.inv_hom_id_assoc, Equiv.coe_fn_mk, comp_app, id_obj,
    rightUnitor_inv_app, Functor.whiskerLeft_app, Functor.whiskerRight_app, map_id,
    associator_inv_app, associator_hom_app, Functor.comp_map, rightUnitor_hom_app,
    leftUnitor_hom_app, comp_id, id_comp, whiskerLeft_twice, Iso.hom_inv_id_assoc]