Lemma tensor_iso_equiv_tensor_eq {C : univalent_category} (TC TD : tensor C) : tensor_eq TC TD ≃ tensor_iso TC TD. Proof. use weqtotal2. - apply weqonsecfibers ; intro x ; apply weqonsecfibers ; intro y. use weq_iso. + apply idtoiso. + apply isotoid. apply univalent_category_is_univalent. + intro. apply isotoid_idtoiso. + intro. apply idtoiso_isotoid. - intro iso_tob. repeat (apply weqonsecfibers ; intro). use weq_iso. + intro p. cbn in p. rewrite (! eq_idtoiso_idtomor _ _ _) in p. rewrite (! eq_idtoiso_idtomor _ _ _) in p. apply p. + intro p. etrans. { apply cancel_postcomposition. apply (! eq_idtoiso_idtomor _ _ _). } etrans. { apply p. } apply cancel_precomposition. apply eq_idtoiso_idtomor. + intro ; apply homset_property. + intro ; apply homset_property. 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.