Definition is_invertible_to_is_disp_invertible {x y : B} {f g : x --> y} {α : f ==> g} (Hα : is_invertible_2cell α) {hx : disp_map_bicat_to_disp_bicat x} {hy : disp_map_bicat_to_disp_bicat y} {hf : hx -->[ f ] hy} {hg : hx -->[ g ] hy} (αα : hf ==>[ α ] hg) (Hαα : is_invertible_2cell (pr1 αα)) : is_disp_invertible_2cell Hα αα. Proof. simple refine (_ ,, _ ,, _). - use make_disp_map_bicat_cell. + exact (Hαα^-1). + abstract (use vcomp_move_R_Mp ; [ is_iso | ] ; cbn ; rewrite !vassocl ; rewrite (pr2 αα) ; rewrite !vassocr ; rewrite lwhisker_vcomp ; rewrite vcomp_linv ; rewrite lwhisker_id2 ; rewrite id2_left ; apply idpath). - abstract (use subtypePath ; [ intro ; apply cellset_property | ] ; cbn ; rewrite transportb_disp_map_bicat_cell ; cbn ; apply vcomp_rinv). - abstract (use subtypePath ; [ intro ; apply cellset_property | ] ; cbn ; rewrite transportb_disp_map_bicat_cell ; cbn ; apply vcomp_linv). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.StreetFibration. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Logic.DisplayMapBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Codomain. Require Import UniMath.Bicategories.Limits.Pullbacks.