Definition disp_nat_z_iso_to_adjequiv (τ : disp_nat_z_iso χ₁ χ₂ (nat_z_iso_id (functor_identity C))) : disp_adjoint_equivalence (D := disp_bicat_comp_cat) (internal_adjoint_equivalence_identity C) χ₁ χ₂. Proof. simple refine (_ ,, ((_ ,, (_ ,, _)) ,, ((_ ,, _) ,, (_ ,, _)))). - exact (disp_nat_z_iso_adjequiv_map τ). - exact (disp_nat_z_iso_adjequiv_inv τ). - abstract (intros x xx ; cbn ; unfold comprehension_nat_trans_mor ; cbn ; rewrite !id_left ; refine (comprehension_functor_mor_id _ _ @ _) ; refine (!_) ; refine (maponpaths pr1 (disp_nat_z_iso_iso_inv τ x xx) @ _) ; rewrite transportb_cod_disp ; apply idpath). - abstract (intros x xx ; cbn ; unfold comprehension_nat_trans_mor ; cbn ; rewrite !id_left ; refine (maponpaths (λ z, _ · z) (comprehension_functor_mor_id _ _) @ _) ; rewrite id_right ; refine (maponpaths pr1 (disp_nat_z_iso_inv_iso τ x xx) @ _) ; rewrite transportb_cod_disp ; apply idpath). - apply disp_2cells_isaprop_disp_bicat_comp_cat. - apply disp_2cells_isaprop_disp_bicat_comp_cat. - apply disp_locally_groupoid_disp_bicat_comp_cat. - apply disp_locally_groupoid_disp_bicat_comp_cat. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain. Require Import UniMath.CategoryTheory.DisplayedCats.Codomain.CodFunctor. Require Import UniMath.CategoryTheory.DisplayedCats.DisplayedFunctorEq. Require Import UniMath.CategoryTheory.DisplayedCats.ComprehensionC. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.DispCatTerminal. Require Import UniMath.Bicategories.ComprehensionCat.BicatOfCompCat.FibTerminal. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat.