Lemma dirprod_disp_cat_is_univalent : is_univalent_disp D1 → is_univalent_disp D2 → is_univalent_disp dirprod_disp_cat. Proof. intros HD1 HD2. apply is_univalent_disp_from_fibers. intros x xx xx'. use isweqhomot. - apply z_iso_disp_aux_weq. + apply is_univalent_in_fibers_from_univalent_disp. apply HD1. + apply is_univalent_in_fibers_from_univalent_disp. apply HD2. - intros p. induction p. cbn. apply (@eq_z_iso_disp _ dirprod_disp_cat). reflexivity. - apply z_iso_disp_aux_weq. Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence.