Definition is_disp_functor_cat_z_iso_if_pointwise_z_iso (x y : FunctorsC'C) (f : z_iso x y) (xx : disp_functor_cat x) (yy : disp_functor_cat y) (FF : xx -->[ f ] yy) (H : forall x' (xx' : D' x') , is_z_iso_disp (pointwise_z_iso_from_nat_z_iso f _ ) (pr1 FF _ xx' )) : is_z_iso_disp f FF. Proof. use tpair. - exact (inv_disp_from_pointwise_z_iso _ _ _ _ _ FF H). - abstract ( split; apply disp_nat_trans_eq; intros c' xx'; refine (_ @ !disp_nat_trans_transportf _ _ _ _ _ _ _ _ _ _); [ refine (z_iso_disp_after_inv_mor (H _ _) @ _) | refine (inv_mor_after_z_iso_disp (H _ _) @ _) ]; apply (maponpaths (λ x, transportf _ x _)); apply homset_property ). 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.FunctorCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos.