Definition inv2cell_weq_disp_adj_equiv_disp_end (HB : is_univalent_2_1 B) {x : B} (ex ey : disp_end x) : invertible_2cell ex ey ≃ disp_adjoint_equivalence (internal_adjoint_equivalence_identity x) ex ey. Proof. use make_weq. - exact inv2cell_to_disp_adj_equiv_disp_end. - use isweq_iso. + exact disp_adj_equiv_to_inv2cell_disp_end. + abstract (intros τ ; use subtypePath ; [ intro ; apply isaprop_is_invertible_2cell | ] ; cbn ; rewrite !vassocr ; rewrite linvunitor_lunitor ; rewrite id2_left ; rewrite !vassocl ; rewrite rinvunitor_runitor ; apply id2_right). + abstract (intros τ ; use subtypePath ; [ intro ; use (isaprop_disp_left_adjoint_equivalence _ _ HB) ; apply disp_univalent_2_1_disp_end | ] ; cbn ; rewrite !vassocr ; rewrite lunitor_linvunitor ; rewrite id2_left ; rewrite !vassocl ; rewrite runitor_rinvunitor ; apply id2_right). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.