Lemma is_univalent_disp_from_is_univalent_fiber {C : category} (D : disp_cat C) : (∏ (c : C), is_univalent D[{c}]) → is_univalent_disp D. Proof. intro H. apply is_univalent_disp_from_fibers. intros c xx xx'. specialize (H c). set (w := make_weq _ (H xx xx')). set (w' := weqcomp w (z_iso_disp_z_iso_fiber D _ xx xx')). apply (weqhomot _ w'). intro e. induction e. apply eq_z_iso_disp. apply idpath. 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.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Total.