-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Braided.Reflection (hypothesis)
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
⊢ IsIso (X := obj (obj yoneda (obj.{v_2, v_2, u_2, u_2} (ihom ((L ⋙ R).obj d)) (R.obj c))) (Opposite.op d')) (Y :=
      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'))
      (NatTrans.app (map yoneda (NatTrans.app (pre (adj.unit.app d)) (R.obj c))) (Opposite.op d')) ↔
    IsIso (X := obj (obj coyoneda (Opposite.op (L.obj (tensorObj.{v_2, u_2} (𝒞 := inst_1) ((L ⋙ R).obj d) d')))) c)
      (NatTrans.app (map coyoneda (Quiver.Hom.op (L.map (adj.unit.app d ▷ d')))) c)
have 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)))
Tactic: have w₁ : ((coyoneda.map (L.map (adj.unit.app d ▷ d')).op).app c) =
        (adj.homEquiv _ _).symm ∘
          (coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c) ∘
            (adj.homEquiv _ _) := by ext; simp