Lemma cat_not_a_two_cat : ¬ (is_strict_bicat bicat_of_univ_cats). Proof. intro H. pose (maponpaths pr1 (eqtohomot (nat_trans_eq_pointwise (maponpaths (λ z, pr1 z) (proofirrelevance _ (@univalent_two_cat_2cells_are_prop (bicat_of_univ_cats ,, H) univalent_cat_is_univalent_2_1 HSET_univalent_category HSET_univalent_category diag_set diag_set) (id2_invertible_2cell _) (nat_z_iso_to_invertible_2cell _ _ swap))) boolset) (true ,, false))) as C. exact (nopathstruetofalse C). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.TwoCategories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.TransportLaws.