Definition is_z_iso_disp_codomain {C : category} {x : C} {fy gz : disp_codomain C x} (φp : fy -->[ identity x ] gz) (H : is_z_isomorphism (pr1 φp)) : is_z_iso_disp (identity_z_iso x) φp. Proof. pose (h := (_ ,, H) : z_iso _ _). simple refine (_ ,, _ ,, _). - refine (inv_from_z_iso h ,, _). abstract (cbn ; use z_iso_inv_on_right ; rewrite id_right ; refine (_ @ !(pr2 φp)) ; 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 h)). 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.