Lemma tensor_iso_disp_eq_equiv {C : bicat_of_univ_cats} (TC TD : bidisp_tensor_disp_bicat C) : tensor_iso TC TD ≃ disp_adjoint_equivalence (idtoiso_2_0 C C (idpath C)) TC TD. Proof. use weq_iso. - intro ti. apply (tensoriso_to_disp_eq TC TD ti). - intro dae. exact (disp_eq_to_tensoriso TC TD dae). - intro i. induction i. use subtypePath. + intro f. repeat (apply impred_isaprop ; intro). apply univalent_category_has_homsets. + repeat (apply funextsec ; intro). use subtypePath. * intro f. apply isaprop_is_z_isomorphism. * repeat (apply funextsec ; intro). apply idpath. - intro dae. use subtypePath. + intro f. apply isaprop_disp_left_adjoint_equivalence. * apply univalent_cat_is_univalent_2_1. * apply bidisp_tensor_disp_bicat_is_locally_univalent. + apply idpath. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.MonoidalCategories.UnivalenceMonCat.CurriedMonoidalCategories. Import TensorNotation. Import TensorNotation.