Lemma tensor_eq''_equiv_tensor_eqi'' {C : univalent_category} (TC TD : tensor C) : tensor_eqi'' TC TD ≃ tensor_eq'' TC TD. Proof. use weq_iso. - intro tei. use tpair. + intros x y. exact (! pr1 tei x y). + intros x1 x2 y1 y2 f g. exact (pr2 tei _ _ _ _ f g). - intro te. use tpair. + intros x y. exact (! pr1 te x y). + intros x1 x2 y1 y2 f g. rewrite pathsinv0inv0. rewrite pathsinv0inv0. apply (pr2 te). - intro te. use total2_paths_b. + repeat (apply funextsec ; intro). apply pathsinv0inv0. + repeat (apply funextsec ; intro). apply homset_property. - intro tei. use total2_paths_b. + repeat (apply funextsec ; intro). apply pathsinv0inv0. + repeat (apply funextsec ; 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.