Definition is_disp_functor_cat_z_iso_iff_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) : (∏ 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. split. - apply is_disp_functor_cat_z_iso_if_pointwise_z_iso. - apply is_pointwise_z_iso_if_is_disp_functor_cat_z_iso. 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.