-- 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)))
tfae_1_to_3 :
  ∀
    (a :
      ∀ (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),
    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
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)))
rw :
  Eq (α :=
    (obj.{v_2, v_2, u_2, u_2} (tensorLeft ((L ⋙ R).obj d)) (Opposite.unop (Opposite.op d')) ⟶ R.obj c) →
      (fun (X : Type v_2) ↦ X)
        (obj (obj yoneda (obj.{v_2, v_2, u_2, u_2} (ihom (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d)) (R.obj c)))
          (Opposite.op d')))
    (Function.comp (β := Opposite.unop (Opposite.op d') ⟶ obj.{v_2, v_2, u_2, u_2} (ihom ((L ⋙ R).obj d)) (R.obj c))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app (map yoneda (NatTrans.app (pre (adj.unit.app d)) (R.obj c))) (Opposite.op d'))) :
        (a : (fun (X : Type v_2) ↦ X) ((yoneda.obj ((ihom ((L ⋙ R).obj d)).obj (R.obj c))).obj (Opposite.op d'))) →
          (fun (X : Type v_2) ↦ X) ((yoneda.obj ((ihom ((𝟭 D).obj d)).obj (R.obj c))).obj (Opposite.op d')))
      (⇑(Adjunction.homEquiv (ihom.adjunction ((L ⋙ R).obj d)) (Opposite.unop (Opposite.op d')) (R.obj c)) :
        (a : (tensorLeft ((L ⋙ R).obj d)).obj (Opposite.unop (Opposite.op d')) ⟶ R.obj c) →
          Opposite.unop (Opposite.op d') ⟶ (ihom ((L ⋙ R).obj d)).obj (R.obj c)))
    (Function.comp (β := obj.{v_2, v_2, u_2, u_2} (tensorLeft d) d' ⟶ R.obj c)
      (⇑(Adjunction.homEquiv (ihom.adjunction d) d' (R.obj c)) :
        (a : (tensorLeft d).obj d' ⟶ R.obj c) → d' ⟶ (ihom d).obj (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))))
⊢ Eq (α :=
    (a :
        (fun (X : Type v_2) ↦ X)
          (obj (obj yoneda (obj.{v_2, v_2, u_2, u_2} (ihom ((L ⋙ R).obj d)) (R.obj c))) (Opposite.op d'))) →
      (fun (X : Type v_2) ↦ X)
        (obj (obj yoneda (obj.{v_2, v_2, u_2, u_2} (ihom (obj.{v_2, v_2, u_2, u_2} (𝟭 D) d)) (R.obj c)))
          (Opposite.op d')))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app (map yoneda (NatTrans.app (pre (adj.unit.app d)) (R.obj c))) (Opposite.op d'))) :
      (a : (fun (X : Type v_2) ↦ X) ((yoneda.obj ((ihom ((L ⋙ R).obj d)).obj (R.obj c))).obj (Opposite.op d'))) →
        (fun (X : Type v_2) ↦ X) ((yoneda.obj ((ihom ((𝟭 D).obj d)).obj (R.obj c))).obj (Opposite.op d')))
    (Function.comp (β := obj.{v_2, v_2, u_2, u_2} (tensorLeft d) d' ⟶ R.obj c)
      (⇑(Adjunction.homEquiv (ihom.adjunction d) d' (R.obj c)) :
        (a : (tensorLeft d).obj d' ⟶ R.obj c) → d' ⟶ (ihom d).obj (R.obj 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)))
        (⇑(Equiv.symm (Adjunction.homEquiv (ihom.adjunction ((L ⋙ R).obj d)) d' (R.obj c))) :
          (a : d' ⟶ (ihom ((L ⋙ R).obj d)).obj (R.obj c)) → (tensorLeft ((L ⋙ R).obj d)).obj d' ⟶ R.obj c)))

Tactic:
rw [← Function.comp_assoc, ((ihom.adjunction ((L ⋙ R).obj d)).homEquiv _ _).eq_comp_symm]

Premises used: CategoryTheory.Functor.comp, Equiv.eq_comp_symm, CategoryTheory.ihom.adjunction, CategoryTheory.Adjunction.homEquiv, CategoryTheory.Functor.obj, Function.comp_assoc