Lemma inv_mor_total_z_iso {C : category} {D : disp_cat C} {xx yy : total_category D} (f : z_iso (pr1 xx) (pr1 yy)) (ff : z_iso_disp f (pr2 xx) (pr2 yy)) : inv_from_z_iso (total_z_iso f ff) = (inv_from_z_iso f,, inv_mor_disp_from_z_iso ff). Proof. apply cancel_precomposition_z_iso with (total_z_iso f ff). etrans. apply z_iso_inv_after_z_iso. apply pathsinv0. use total2_paths_f; cbn. - apply z_iso_inv_after_z_iso. - etrans. apply maponpaths, inv_mor_after_z_iso_disp. apply transportfbinv. Qed.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.AxiomOfChoice. 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.