Lemma z_iso_disp_prod_weq (x : C) (xx1 xx1' : D1 x) (xx2 xx2' : D2 x) : @z_iso_disp _ dirprod_disp_cat _ _ (identity_z_iso x) (xx1,, xx2) (xx1',, xx2') ≃ (z_iso_disp (identity_z_iso x) xx1 xx1') × (z_iso_disp (identity_z_iso x) xx2 xx2'). Proof. exists (z_iso_disp_prod1 xx1 xx1' xx2 xx2'). use isweq_iso. - apply z_iso_disp_prod2. - apply z_iso_disp_prod21. - apply z_iso_disp_prod12. 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.