Definition is_univalent_2_1_weq_local_univ (C : bicat) : is_univalent_2_1 C ≃ (∏ (X Y : C), is_univalent (hom X Y)). Proof. use weqiff. - split. + exact is_univ_hom. + exact is_univalent_2_1_if_hom_is_univ. - apply isaprop_is_univalent_2_1. - use impred ; intro. use impred ; intro. apply isaprop_is_univalent. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Groupoids. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.BicategoryLaws.