proposition: 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: 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.
imports: UniMath.Foundations.Sets UniMath.MoreFoundations.AxiomOfChoice UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence