Definition is_z_iso_disp_transportf_fun_eq {C : category} {D : disp_cat C} {x y : C} {f g : z_iso x y} {xx : D x} {yy : D y} (ff : xx -->[ f ] yy) (p : pr1 f = pr1 g) (Hff : is_z_iso_disp f ff) : is_z_iso_disp g (transportf (λ z, _ -->[ z ] _) p ff). Proof. simple refine (_ ,, _ ,, _). - refine (transportf (λ z, _ -->[ z ] _) _ (inv_mor_disp_from_z_iso Hff)). abstract (induction f as [ f Hf ] ; induction g as [ g Hg ] ; cbn in * ; induction p ; do 2 apply maponpaths ; apply isaprop_is_z_isomorphism). - abstract (induction f as [ f Hf ] ; induction g as [ g Hg ] ; cbn in * ; induction p ; cbn ; rewrite mor_disp_transportf_postwhisker ; refine (maponpaths _ (z_iso_disp_after_inv_mor Hff) @ _) ; unfold transportb ; rewrite transport_f_f ; apply maponpaths_2 ; apply homset_property). - abstract (induction f as [ f Hf ] ; induction g as [ g Hg ] ; cbn in * ; induction p ; cbn ; rewrite mor_disp_transportf_prewhisker ; refine (maponpaths _ (inv_mor_after_z_iso_disp Hff) @ _) ; unfold transportb ; rewrite transport_f_f ; apply maponpaths_2 ; apply homset_property). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core.