Definition is_z_iso_disp_codomain' {C : category} {x y : C} (h : z_iso x y) {fz : disp_codomain C x} {fz' : disp_codomain C y} (φp : fz -->[ h ] fz') (H : is_z_isomorphism (pr1 φp)) : is_z_iso_disp h φp. Proof. pose (l := (_ ,, H) : z_iso _ _). simple refine (_ ,, _ ,, _). - refine (inv_from_z_iso l ,, _). abstract (cbn ; use z_iso_inv_on_right ; rewrite !assoc ; refine (_ @ maponpaths (λ z, z · _) (!(pr2 φp))) ; rewrite !assoc' ; rewrite z_iso_inv_after_z_iso ; rewrite id_right ; apply idpath). - abstract (use eq_cod_mor ; rewrite transportb_cod_disp ; apply z_iso_after_z_iso_inv). - abstract (use eq_cod_mor ; rewrite transportb_cod_disp ; apply (z_iso_inv_after_z_iso l)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations.