-- 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)))))
rw :
  IsIso.{v_2, v_2 + 1} (X := tensorObj.{v_2, u_2} (𝒞 := inst_1) (R.obj (L.obj d)) d' ⟶ R.obj c) (Y :=
    tensorObj.{v_2, u_2} (𝒞 := inst_1) 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)))) :
      obj (obj yoneda (R.obj c)) (tensorObj.{v_2, u_2} (𝒞 := opposite) (Opposite.op ((L ⋙ R).1 d)) (Opposite.op d')) ⟶
        obj (obj yoneda (R.obj c))
          (tensorObj.{v_2, u_2} (𝒞 := opposite) (Opposite.op ((𝟭 D : Functor.{v_2, v_2, u_2, u_2} D D).1 d))
            (Opposite.op d')))
⊢ Function.Bijective (α := tensorObj.{v_2, u_2} (𝒞 := inst_1) (R.obj (L.obj d)) d' ⟶ R.obj c) (β :=
    tensorObj.{v_2, u_2} (𝒞 := inst_1) d d' ⟶ R.obj c)
    (⇑(@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
          (NatTrans.app (map coyoneda (Quiver.Hom.op (adj.unit.app d ▷ d'))) (R.obj c))) :
      (a : R.obj (L.obj d) ⊗ d' ⟶ R.obj c) → d ⊗ d' ⟶ R.obj c)

Tactic:
rw [w₂, ← isIso_iff_bijective]

Premises used: CategoryTheory.isIso_iff_bijective