Lemma isaprop_is_z_iso_disp {C : category} {D : disp_cat C} {x y : C} (f : z_iso x y) {xx : D x} {yy} (ff : xx -->[f] yy) : isaprop (is_z_iso_disp f ff). Proof. apply invproofirrelevance; intros i i'. apply subtypePath. - intros gg. apply isaprop_is_disp_inverse. - destruct i as [gg [gf fg]], i' as [gg' [gf' fg']]; simpl. etrans. eapply pathsinv0, transportfbinv. etrans. apply maponpaths, @pathsinv0, id_right_disp. etrans. apply maponpaths, maponpaths. etrans. eapply pathsinv0, transportfbinv. apply maponpaths, @pathsinv0, fg'. etrans. apply maponpaths, mor_disp_transportf_prewhisker. etrans. apply transport_f_f. etrans. apply maponpaths, assoc_disp. etrans. apply transport_f_f. etrans. apply maponpaths, maponpaths_2, gf. etrans. apply maponpaths, mor_disp_transportf_postwhisker. etrans. apply transport_f_f. etrans. apply maponpaths, id_left_disp. etrans. apply transport_f_f. use (@maponpaths_2 _ _ _ (transportf _) _ (idpath _)). apply homset_property. Qed.
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.