Definition disp_cell_unit_bicat_left_adjoint_equivalence_weq {a b : C} (f : adjoint_equivalence a b) {aa : disp_cell_unit_bicat a} {bb : disp_cell_unit_bicat b} (ff : aa -->[ f] bb) : (bb -->[ left_adjoint_right_adjoint f] aa) ≃ disp_left_adjoint_equivalence f ff. Proof. refine (disp_cell_unit_bicat_left_adjoint_equivalence f ff ,, _). use isweq_iso. - exact (disp_cell_unit_bicat_left_adjoint_equivalence_inv f ff). - reflexivity. - intros y. use subtypePath. + intro. do 2 apply isapropdirprod. * apply disp_cell_unit_bicat_disp_cellset. * apply disp_cell_unit_bicat_disp_cellset. * apply isaprop_is_disp_invertible_2cell. * apply isaprop_is_disp_invertible_2cell. + cbn. unfold disp_cell_unit_bicat_left_adjoint_equivalence_inv. use total2_paths_b. * reflexivity. * use total2_paths_b ; apply isapropunit. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.Initial. Require Import UniMath.Bicategories.Core.Examples.Final. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.