Definition weq_fam_global (HB_2_1 : is_univalent_2_1 B) (HD : arrow_subbicat_props D) {x : B} (hx hy : disp_map_bicat_to_disp_bicat x) (p : pr1 hx = pr1 hy) : (transportf (λ z, ∑ (h : z --> x), pred_ob D h) p (pr2 hx) = pr2 hy) ≃ invertible_2cell (pr12 hx) (pr1 (idtoiso_2_0 (pr1 hx) (pr1 hy) p) · pr12 hy). Proof. induction hx as [ hx₁ [ hx₂ hx₃ ] ]. induction hy as [ hy₁ [ hy₂ hy₃ ] ]. cbn in *. induction p ; cbn. refine (_ ∘ path_sigma_hprop _ _ _ _)%weq. - apply HD. - exact (cod_1cell_path_help hx₂ hy₂ ∘ make_weq _ (HB_2_1 _ _ _ _))%weq. 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.CategoryTheory.DisplayedCats.StreetFibration. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.InternalStreetFibration. Require Import UniMath.Bicategories.Morphisms.InternalStreetOpFibration. Require Import UniMath.Bicategories.Logic.DisplayMapBicat. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Codomain. Require Import UniMath.Bicategories.Limits.Pullbacks.