Definition adjequiv_to_disp_nat_z_iso (τ : disp_adjoint_equivalence (D := disp_bicat_comp_cat) (internal_adjoint_equivalence_identity C) χ₁ χ₂) : disp_nat_z_iso χ₁ χ₂ (nat_z_iso_id (functor_identity C)). Proof. simple refine ((_ ,, _) ,, _). - exact (λ x xx, pr11 τ x xx). - abstract (intros x y f xx yy ff ; use subtypePath ; [ intro ; apply homset_property | ] ; rewrite transportb_cod_disp ; cbn ; pose (p := maponpaths pr1 (disp_nat_trans_ax (pr1 τ) ff)) ; rewrite transportb_cod_disp in p ; exact p). - intros x xx ; cbn. simple refine (_ ,, (_ ,, _)). + exact (pr1 (pr112 τ) x xx). + abstract (use subtypePath ; [ intro ; apply homset_property | ] ; rewrite transportb_cod_disp ; cbn ; pose (p := pr2 (pr212 τ) x xx) ; cbn in p ; rewrite !id_left in p ; refine (_ @ p) ; refine (!(id_right _) @ _) ; apply maponpaths ; refine (!_) ; apply comprehension_functor_mor_id). + abstract (use subtypePath ; [ intro ; apply homset_property | ] ; rewrite transportb_cod_disp ; cbn ; pose (p := pr1 (pr212 τ) x xx) ; cbn in p ; rewrite !id_left in p ; refine (!p @ _) ; apply comprehension_functor_mor_id). 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.