Definition map1cells_disp_univalent_2_0 (HD_2_1 : is_univalent_2_1 D) : disp_univalent_2_0 map1cells_disp_bicat. Proof. apply fiberwise_univalent_2_0_to_disp_univalent_2_0. intros F₀ F₁ F₁'. use weqhomot. - simple refine (_ ∘ make_weq _ (isweqtoforallpaths _ _ _))%weq. simple refine (_ ∘ weqonsecfibers _ _ _)%weq. + exact (λ X, ∏ (Y : C) (f : X --> Y), F₁ X Y f = F₁' X Y f). + intro X ; cbn. simple refine (_ ∘ make_weq _ (isweqtoforallpaths _ _ _))%weq. simple refine (weqonsecfibers _ _ _)%weq. intro Y ; cbn. simple refine (_ ∘ make_weq _ (isweqtoforallpaths _ _ _))%weq. apply idweq. + refine (_ ∘ weqonsecfibers _ _ _)%weq. * intro X ; cbn. refine (weqonsecfibers _ _ _). intro Y ; cbn. simple refine (weqonsecfibers _ _ _). -- exact (λ f, invertible_2cell (F₁ X Y f) (F₁' X Y f)). -- intro f ; cbn. exact (make_weq (idtoiso_2_1 (F₁ X Y f) (F₁' X Y f)) (HD_2_1 _ _ _ _)). * exact (all_invertible_2cell_is_disp_adjoint_equivalence HD_2_1 F₀ F₁ F₁'). - intro p. induction p. apply subtypePath. { intro. apply isaprop_disp_left_adjoint_equivalence. + exact (ps_base_is_univalent_2_1 _ _ HD_2_1). + apply map1cells_disp_univalent_2_1. } apply funextsec ; intro X. apply funextsec ; intro Y. apply funextsec ; intro f. apply subtypePath. { intro ; apply isaprop_is_invertible_2cell. } cbn. rewrite id2_right. reflexivity. 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.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Display.Base. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. 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.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.