proposition: Lemma disp_functor_on_z_iso_disp_aux1 {C C'} {F} {D : disp_cat C} {D' : disp_cat C'} (FF : disp_functor F D D') {x y} {xx : D x} {yy} {f : z_iso x y} (ff : xx -->[f] yy) (Hff : is_z_iso_disp f ff) : transportf _ (functor_on_inv_from_z_iso F f) (♯ FF (inv_mor_disp_from_z_iso Hff)) ;; ♯ FF ff = transportb _ (z_iso_after_z_iso_inv _) (id_disp _).
proof: Proof. etrans. apply mor_disp_transportf_postwhisker. etrans. apply maponpaths, @pathsinv0, disp_functor_comp_var. etrans. apply transport_f_f. etrans. apply maponpaths, maponpaths, z_iso_disp_after_inv_mor. etrans. apply maponpaths, disp_functor_transportf. etrans. apply transport_f_f. etrans. apply maponpaths, disp_functor_id. etrans. apply transport_f_b. unfold transportb. apply maponpaths_2, homset_property. Qed.
imports: UniMath.Foundations.Sets UniMath.MoreFoundations.PartA UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos