-- Module: Mathlib.CategoryTheory.Monoidal.Braided.Reflection
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : MonoidalCategory.{v_2, u_2} D
inst_3 : SymmetricCategory.{v_2, u_2} D
inst_4 : MonoidalClosed.{v_2, u_2} D
R : Functor.{v_1, v_2, u_1, u_2} C D
inst_5 : R.Faithful
inst_6 : R.Full
L : Functor.{v_2, v_1, u_2, u_1} D C
adj : L ⊣ R
tfae_3_to_4 :
  ∀
    (a :
      ∀ (d d' : D),
        IsIso (X := L.obj (tensorObj.{v_2, u_2} (𝒞 := inst_1) (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d) d'))
          (L.map (adj.unit.app d ▷ d')))
    (d d' : D),
    IsIso (X :=
      L.obj (tensorObj.{v_2, u_2} (𝒞 := inst_1) (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d) (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d')))
      (L.map (adj.unit.app d ⊗ₘ adj.unit.app d'))
tfae_4_to_1 :
  ∀
    (a :
      ∀ (d d' : D),
        IsIso (X :=
          L.obj
            (tensorObj.{v_2, u_2} (𝒞 := inst_1) (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d) (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d')))
          (L.map (adj.unit.app d ⊗ₘ adj.unit.app d')))
    (c : C) (d : D),
    IsIso (X := obj.{v_2, v_2, u_2, u_2} (𝟭 D) (obj.{v_2, v_2, u_2, u_2} (ihom d) (R.obj c)))
      (adj.unit.app (obj.{v_2, v_2, u_2, u_2} (ihom d) (R.obj c)))
h :
  ∀ (c : C) (d : D),
    IsIso (X := obj.{v_2, v_2, u_2, u_2} (𝟭 D) (obj.{v_2, v_2, u_2, u_2} (ihom d) (R.obj c)))
      (adj.unit.app (obj.{v_2, v_2, u_2, u_2} (ihom d) (R.obj c)))
d d' : D
c : C
w₁ :
  Eq (α :=
    (a :
        (fun (X : Type v_1) ↦ X)
          (obj (obj coyoneda (Opposite.op (L.obj (tensorObj.{v_2, u_2} (𝒞 := inst_1) ((L ⋙ R).obj d) d')))) c)) →
      (fun (X : Type v_1) ↦ X)
        (obj
          (obj coyoneda
            (Opposite.op (L.obj (tensorObj.{v_2, u_2} (𝒞 := inst_1) (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d) d'))))
          c))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app (map coyoneda (Quiver.Hom.op (L.map (adj.unit.app d ▷ d')))) c)) :
      (a : (fun (X : Type v_1) ↦ X) ((coyoneda.obj (Opposite.op (L.obj ((L ⋙ R).obj d ⊗ d')))).obj c)) →
        (fun (X : Type v_1) ↦ X) ((coyoneda.obj (Opposite.op (L.obj ((𝟭 D).obj d ⊗ d')))).obj c))
    (Function.comp (β :=
      Opposite.unop (Opposite.op (tensorObj.{v_2, u_2} (𝒞 := inst_1) (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d) d')) ⟶ R.obj c)
      (⇑(Equiv.symm
            (adj.homEquiv
              (Opposite.unop (Opposite.op (tensorObj.{v_2, u_2} (𝒞 := inst_1) (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d) d')))
              c)) :
        (a : Opposite.unop (Opposite.op ((𝟭 D).obj d ⊗ d')) ⟶ R.obj c) →
          L.obj (Opposite.unop (Opposite.op ((𝟭 D).obj d ⊗ d'))) ⟶ c)
      (Function.comp (β :=
        (fun (X : Type v_2) ↦ X)
          (obj (obj coyoneda (Opposite.op (tensorObj.{v_2, u_2} (𝒞 := inst_1) ((L ⋙ R).obj d) d'))) (R.obj c)))
        (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app (map coyoneda (Quiver.Hom.op (adj.unit.app d ▷ d'))) (R.obj c))) :
          (a : (fun (X : Type v_2) ↦ X) ((coyoneda.obj (Opposite.op ((L ⋙ R).obj d ⊗ d'))).obj (R.obj c))) →
            (fun (X : Type v_2) ↦ X) ((coyoneda.obj (Opposite.op ((𝟭 D).obj d ⊗ d'))).obj (R.obj c)))
        (⇑(adj.homEquiv (Opposite.unop (Opposite.op (tensorObj.{v_2, u_2} (𝒞 := inst_1) ((L ⋙ R).obj d) d'))) c) :
          (a : L.obj (Opposite.unop (Opposite.op ((L ⋙ R).obj d ⊗ d'))) ⟶ c) →
            Opposite.unop (Opposite.op ((L ⋙ R).obj d ⊗ d')) ⟶ R.obj c)))
w₂ :
  Eq (α :=
    obj (obj coyoneda (Opposite.op (tensorObj.{v_2, u_2} (𝒞 := inst_1) ((L ⋙ R).obj d) d'))) (R.obj c) ⟶
      obj (obj coyoneda (Opposite.op (tensorObj.{v_2, u_2} (𝒞 := inst_1) (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d) d')))
        (R.obj c))
    (NatTrans.app (map coyoneda (Quiver.Hom.op (adj.unit.app d ▷ d'))) (R.obj c))
    (Iso.hom (mapIso (obj yoneda (R.obj c)) (β_ (Opposite.op ((L ⋙ R).1 d)) (Opposite.op d'))) ≫
      NatTrans.app (map coyoneda (Quiver.Hom.op (d' ◁ adj.unit.app d))) (R.obj c) ≫
        Iso.hom
          (mapIso (obj yoneda (R.obj c))
            (β_ (Opposite.op d') (Opposite.op ((𝟭 D : Functor.{v_2, v_2, u_2, u_2} D D).1 d)))))
w₃ :
  Eq (α :=
    (a :
        (fun (X : Type v_2) ↦ X)
          (obj (obj coyoneda (Opposite.op (tensorObj.{v_2, u_2} (𝒞 := inst_1) d' ((L ⋙ R).obj d)))) (R.obj c))) →
      (fun (X : Type v_2) ↦ X)
        (obj (obj coyoneda (Opposite.op (tensorObj.{v_2, u_2} (𝒞 := inst_1) d' (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d))))
          (R.obj c)))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app (map coyoneda (Quiver.Hom.op (d' ◁ adj.unit.app d))) (R.obj c))) :
      (a : (fun (X : Type v_2) ↦ X) ((coyoneda.obj (Opposite.op (d' ⊗ (L ⋙ R).obj d))).obj (R.obj c))) →
        (fun (X : Type v_2) ↦ X) ((coyoneda.obj (Opposite.op (d' ⊗ (𝟭 D).obj d))).obj (R.obj c)))
    (Function.comp (β :=
      Opposite.unop (Opposite.op (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d)) ⟶ obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c))
      (⇑(Equiv.symm
            (Adjunction.homEquiv (ihom.adjunction d') (Opposite.unop (Opposite.op (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d)))
              (R.obj c))) :
        (a : Opposite.unop (Opposite.op ((𝟭 D).obj d)) ⟶ (ihom d').obj (R.obj c)) →
          (tensorLeft d').obj (Opposite.unop (Opposite.op ((𝟭 D).obj d))) ⟶ R.obj c)
      (Function.comp (β :=
        (fun (X : Type v_2) ↦ X)
          (obj (obj coyoneda (Opposite.op ((L ⋙ R).obj d))) (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c))))
        (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app (map coyoneda (adj.unit.app d).op) (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c)))) :
          (a : (fun (X : Type v_2) ↦ X) ((coyoneda.obj (Opposite.op ((L ⋙ R).obj d))).obj ((ihom d').obj (R.obj c)))) →
            (fun (X : Type v_2) ↦ X) ((coyoneda.obj (Opposite.op ((𝟭 D).obj d))).obj ((ihom d').obj (R.obj c))))
        (⇑(Adjunction.homEquiv (ihom.adjunction d') (Opposite.unop (Opposite.op ((L ⋙ R).obj d))) (R.obj c)) :
          (a : (tensorLeft d').obj (Opposite.unop (Opposite.op ((L ⋙ R).obj d))) ⟶ R.obj c) →
            Opposite.unop (Opposite.op ((L ⋙ R).obj d)) ⟶ (ihom d').obj (R.obj c))))
w₄ :
  Eq (α :=
    obj (obj coyoneda (Opposite.op ((L ⋙ R).obj d))) (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c)) ⟶
      obj (obj coyoneda (Opposite.op d)) ((L ⋙ R).obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c))))
    (NatTrans.app (map coyoneda (adj.unit.app d).op) (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c)) ≫
      map (obj coyoneda (Opposite.op d)) (adj.unit.app (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c))))
    (map (obj coyoneda (Opposite.op ((L ⋙ R).obj d))) (adj.unit.app (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c))) ≫
      NatTrans.app (map coyoneda (adj.unit.app d).op) ((L ⋙ R).obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c))))
f : obj (obj coyoneda (Opposite.op ((L ⋙ R).obj d))) ((L ⋙ R).obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c)))
out :
  R.map (L.map (adj.unit.app d)) ≫
      R.map (L.map f) ≫ R.map (adj.counit.app (L.obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c)))) =
    f
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app (map coyoneda (adj.unit.app d).op) ((L ⋙ R).obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c))) ≫
          ↾fun (f : obj (obj coyoneda (Opposite.op d)) ((L ⋙ R).obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c)))) ↦
            R.map
              (((Equiv.symm (adj.homEquiv d (L.obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c)))) :
                    Equiv.{v_2 + 1, v_1 + 1} (d ⟶ R.obj (L.obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c))))
                      (L.obj d ⟶ L.obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c)))) :
                  (a : d ⟶ R.obj (L.obj ((ihom d').obj (R.obj c)))) → L.obj d ⟶ L.obj ((ihom d').obj (R.obj c)))
                f)))
      f =
    (ConcreteCategory.hom (FC := TypeCat.Fun)
          (𝟙
            (obj (obj coyoneda (Opposite.op ((L ⋙ R).obj d)))
              ((L ⋙ R).obj (obj.{v_2, v_2, u_2, u_2} (ihom d') (R.obj c)))))).toFun
      f

Tactic:
simp only [comp_obj, flip_obj_obj, yoneda_obj_obj, id_obj, flip_map_app, yoneda_obj_map,
      Quiver.Hom.unop_op, Adjunction.homEquiv_counit, map_comp, TypeCat.Fun.toFun_apply, comp_apply,
      ConcreteCategory.hom_ofHom, TypeCat.Fun.coe_mk, Category.assoc, id_apply]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Category.assoc, Quiver.Hom.unop_op, TypeCat.Fun.coe_mk, CategoryTheory.yoneda_obj_obj, CategoryTheory.yoneda_obj_map, TypeCat.Fun.toFun_apply, CategoryTheory.id_apply, CategoryTheory.comp_apply, CategoryTheory.Functor.flip_obj_obj, CategoryTheory.Functor.id_obj, CategoryTheory.Functor.map_comp, CategoryTheory.Adjunction.homEquiv_counit, CategoryTheory.Functor.flip_map_app, CategoryTheory.ConcreteCategory.hom_ofHom