Definition slice_inv2cell_weq_disp_adj_equiv (HB : is_univalent_2_1 B) {x : B} (f g : slice_disp_bicat x) : invertible_2cell f g ≃ disp_adjoint_equivalence (internal_adjoint_equivalence_identity x) f g. Proof. use make_weq. - exact slice_inv2cell_to_disp_adj_equiv. - use isweq_iso. + exact 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 _ _ HB disp_univalent_2_1_slice) | ] ; 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.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.