Lemma bicatcattensor_disp_prebicat_is_globally_univalent : disp_univalent_2_0 bidisp_tensor_disp_bicat. Proof. intros C D equalcats TC TD. induction equalcats. use weqhomot. - set (i1 := tensor_iso_disp_eq_equiv TC TD). set (i2 := tensor_iso_equiv_tensor_eq TC TD). set (i3 := tensor_eq_equiv_tensor_eq' TC TD). set (i4 := tensor_eq'_equiv_tensor_eq'' TC TD). set (i5 := tensor_eq''_equiv_tensor_eqi'' TC TD). set (i6 := tensor_eqi''_equiv_tensor_eqi''' TC TD). set (i7 := tensor_eqi'''_equiv_eq TC TD). exact ((i1 ∘ i2 ∘ i3 ∘ i4 ∘ i5 ∘ i6 ∘ i7)%weq). - intro p. induction p. use subtypePath. + intro. apply (@isaprop_disp_left_adjoint_equivalence bicat_of_univ_cats bidisp_tensor_disp_bicat). * exact univalent_cat_is_univalent_2_1. * exact bidisp_tensor_disp_bicat_is_locally_univalent. + use total2_paths_b. * repeat (apply funextsec ; intro). apply idpath. * repeat (apply funextsec ; intro). apply homset_property. Qed.
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.