Lemma tensor_eqi''_equiv_tensor_eqi''' {C : univalent_category} (TC TD : tensor C) : tensor_eqi''' TC TD ≃ tensor_eqi'' TC TD. Proof. apply weqfibtototal. intro. repeat (apply weqonsecfibers ; intro). use weq_iso. - intro q. apply pathsinv0. etrans. { apply maponpaths. apply maponpaths. exact (! q). } rewrite (! independent_transportation_commutes _ _ _). rewrite transport_f_f. rewrite pathsinv0r. rewrite idpath_transportf. rewrite transport_f_f. rewrite pathsinv0r. apply idpath_transportf. - intro q. etrans. { apply maponpaths. apply maponpaths. exact q. } rewrite (! independent_transportation_commutes _ _ _). rewrite transport_f_f. rewrite pathsinv0l. rewrite idpath_transportf. rewrite transport_f_f. rewrite pathsinv0l. apply idpath_transportf. - intro q. apply homset_property. - intro q. 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.