Definition inv2_cell_to_disp_adjequiv : disp_adjoint_equivalence (internal_adjoint_equivalence_identity x) fx fy. Proof. use tpair. - exact inv2_cell_to_disp_adjequiv_left_adj. - use tpair. + use tpair. * exact inv2_cell_to_disp_adjequiv_right_adj. * split. ** exact inv2_cell_to_disp_adjequiv_unit. ** exact inv2_cell_to_disp_adjequiv_counit. + abstract (refine ((_ ,, _) ,, (_ ,, _)) ; try apply C ; apply disp_inserter_locally_groupoidal). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. 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.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell.