Definition disp_map_slice_inv2cell_weq_disp_adj_equiv (HB : is_univalent_2 B) (HD : arrow_subbicat_props D) {x : B} (f g : disp_map_slice_disp_bicat x) : invertible_2cell (pr1 f) (pr1 g) ≃ disp_adjoint_equivalence (internal_adjoint_equivalence_identity x) f g. Proof. use make_weq. - exact (disp_map_slice_inv2cell_to_disp_adj_equiv HB). - use isweq_iso. + exact disp_map_slice_disp_adj_equiv_to_inv2cell. + abstract (intros α ; use subtypePath ; [ intro ; apply isaprop_is_invertible_2cell | ] ; cbn ; rewrite !vassocl ; rewrite linvunitor_lunitor ; apply id2_right). + abstract (intros α ; use subtypePath ; [ intro ; apply (isaprop_disp_left_adjoint_equivalence _ _ (pr2 HB) (disp_univalent_2_1_disp_map_slice HD)) | ] ; use pathsdirprod ; [ apply HD | use subtypePath ; [ intro ; apply isaprop_is_invertible_2cell | ] ; cbn ; rewrite !vassocl ; rewrite lunitor_linvunitor ; apply id2_right]). 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.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Discreteness. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.Logic.DisplayMapBicat. Require Import UniMath.Bicategories.Limits.Pullbacks.