Definition isaprop_is_universal_iso_inserter_cone (HB_2_1 : is_univalent_2_1 B) (cone : iso_inserter_cone) : isaprop (is_universal_iso_inserter_cone cone). Proof. use impred ; intro x. use isofhlevelweqf. - exact (@left_adjoint_equivalence bicat_of_univ_cats (univ_hom HB_2_1 x cone) (@univalent_cat_iso_inserter (univ_hom HB_2_1 _ _) (univ_hom HB_2_1 _ _) (post_comp x f) (post_comp x g)) (iso_inserter_cone_functor cone x)). - exact (@adj_equiv_is_equiv_cat (univ_hom HB_2_1 x cone) (@univalent_cat_iso_inserter (univ_hom HB_2_1 _ _) (univ_hom HB_2_1 _ _) (post_comp x f) (post_comp x g)) (iso_inserter_cone_functor cone x)). - apply isaprop_left_adjoint_equivalence. exact univalent_cat_is_univalent_2_1. 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.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful. Require Import UniMath.CategoryTheory.Categories.CatIsoInserter. 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.AdjointUnique. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.DiscreteMorphisms.