Definition disp_univalent_2_0_disp_map_slice (HB : is_univalent_2 B) (HD : arrow_subbicat_props D) : disp_univalent_2_0 disp_map_slice_disp_bicat. Proof. use fiberwise_univalent_2_0_to_disp_univalent_2_0. intros x f g. use weqhomot. - refine (disp_map_slice_inv2cell_weq_disp_adj_equiv HB HD f g ∘ make_weq _ (pr2 HB _ _ (pr1 f) (pr1 g)) ∘ path_sigma_hprop _ _ _ _)%weq. apply HD. - abstract (intro p ; cbn in p ; induction p ; use subtypePath ; [ intro ; apply (isaprop_disp_left_adjoint_equivalence _ _ (pr2 HB) (disp_univalent_2_1_disp_map_slice HD)) | ] ; use pathsdirprod ; [ apply HD | ] ; use subtypePath ; [ intro ; apply isaprop_is_invertible_2cell | ] ; cbn ; apply id2_left). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Discreteness. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.Logic.DisplayMapBicat. Require Import UniMath.Bicategories.Limits.Pullbacks.