Module: Mathlib.CategoryTheory.Monoidal.Braided.Reflection

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : MonoidalCategory D
inst✝³ : SymmetricCategory D
inst✝² : MonoidalClosed D
R : C ⥤ D
inst✝¹ : R.Faithful
inst✝ : R.Full
L : D ⥤ C
adj : L ⊣ R
tfae_3_to_4 :
  (∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))) → ∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ₘ adj.unit.app d'))
tfae_4_to_1 :
  (∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ₘ adj.unit.app d'))) →
    ∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
⊢ [∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c))),
      ∀ (c : C) (d : D), IsIso ((pre (adj.unit.app d)).app (R.obj c)),
      ∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d')),
      ∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ₘ adj.unit.app d'))].TFAE

Tactic:
tfae_have 1 → 3
  | h, d, d' => by
    rw [isIso_iff_isIso_coyoneda_map]
    intro c
    -- `w₁, w₃, w₄` are the three stacked commutative squares in the proof on nLab:
    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
    rw [isIso_iff_bijective, w₁]
    simp only [comp_obj, flip_obj_obj, yoneda_obj_obj, id_obj, op_tensorObj, unop_tensorObj,
      EquivLike.comp_bijective, EquivLike.bijective_comp]
    -- We commute the tensor product using the auxiliary commutative square `w₂`.
    have w₂ : ((coyoneda.map (adj.unit.app d ▷ d').op).app (R.obj c)) =
        ((yoneda.obj (R.obj c)).mapIso (β_ _ _)).hom ≫
          ((coyoneda.map (d' ◁ adj.unit.app d).op).app (R.obj c)) ≫
            ((yoneda.obj (R.obj c)).mapIso (β_ _ _)).hom := by ext; simp
    rw [w₂, ← isIso_iff_bijective]
    suffices IsIso <| (coyoneda.map (d' ◁ adj.unit.app d).op).app (R.obj c) by infer_instance
    have w₃ : ((coyoneda.map (d' ◁ adj.unit.app d).op).app (R.obj c) : _ → _) =
        ((ihom.adjunction d').homEquiv _ _).symm ∘
          ((coyoneda.map (adj.unit.app _).op).app _) ∘ (ihom.adjunction d').homEquiv _ _ := by
      ext
      simp only [id_obj, op_tensorObj, flip_obj_obj, yoneda_obj_obj, unop_tensorObj, comp_obj,
        flip_map_app, Function.comp_apply, Adjunction.homEquiv_unit, Adjunction.homEquiv_counit]
      simp
    rw [isIso_iff_bijective, w₃]
    simp only [comp_obj, op_tensorObj, flip_obj_obj, yoneda_obj_obj, unop_tensorObj, id_obj,
      yoneda_obj_obj, curriedTensor_obj_obj, EquivLike.comp_bijective, EquivLike.bijective_comp]
    have w₄ : (coyoneda.map (adj.unit.app d).op).app ((ihom d').obj (R.obj c)) ≫
        (coyoneda.obj ⟨d⟩).map (adj.unit.app ((ihom d').obj (R.obj c))) =
          (coyoneda.obj ⟨(L ⋙ R).obj d⟩).map (adj.unit.app ((ihom d').obj (R.obj c))) ≫
            (coyoneda.map (adj.unit.app d).op).app _ := by cat_disch
    rw [← isIso_iff_bijective]
    suffices IsIso ((coyoneda.map (adj.unit.app d).op).app ((ihom d').obj (R.obj c)) ≫
        (coyoneda.obj ⟨d⟩).map (adj.unit.app ((ihom d').obj (R.obj c)))) from
      IsIso.of_isIso_comp_right _ ((coyoneda.obj ⟨d⟩).map (adj.unit.app ((ihom d').obj (R.obj c))))
    rw [w₄]
    refine IsIso.comp_isIso' inferInstance ?_
    constructor
    -- We give the inverse of the bottom map in the stack of commutative squares:
    refine ⟨↾fun f ↦ R.map ((adj.homEquiv _ _).symm f), ?_, by ext; simp⟩
    ext f
    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]
    have : f = R.map (R.preimage f) := by simp
    rw [this]
    simp [← map_comp, -map_preimage]

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : MonoidalCategory D
inst✝³ : SymmetricCategory D
inst✝² : MonoidalClosed D
R : C ⥤ D
inst✝¹ : R.Faithful
inst✝ : R.Full
L : D ⥤ C
adj : L ⊣ R
tfae_3_to_4 :
  (∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))) → ∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ₘ adj.unit.app d'))
tfae_4_to_1 :
  (∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ₘ adj.unit.app d'))) →
    ∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))
tfae_1_to_3 :
  (∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c)))) → ∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d'))
⊢ [∀ (c : C) (d : D), IsIso (adj.unit.app ((ihom d).obj (R.obj c))),
      ∀ (c : C) (d : D), IsIso ((pre (adj.unit.app d)).app (R.obj c)),
      ∀ (d d' : D), IsIso (L.map (adj.unit.app d ▷ d')),
      ∀ (d d' : D), IsIso (L.map (adj.unit.app d ⊗ₘ adj.unit.app d'))].TFAE