Lemma functor_tensor_disp_cat_is_univalent : is_univalent_disp (functor_tensor_disp_cat TC TD). Proof. apply is_univalent_disp_from_fibers. intros F pt1 pt2. use isweqimplimpl. - intro i. use total2_paths_f. + apply funextsec ; intro. set (ix := (pr1 i) (pr1 x) (pr2 x)). cbn in ix. rewrite binprod_id in ix. rewrite (functor_id TD) in ix. rewrite id_right in ix. rewrite id_left in ix. exact (! ix). + do 2 (apply funextsec ; intro). repeat (apply impred_isaprop ; intro). apply D. - apply isaset_functor_tensor_disp_cat. - apply isaproptotal2. + intro ; apply Isos.isaprop_is_z_iso_disp. + do 4 intro ; apply isaprop_is_nat_trans_tensor. 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.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored.