Lemma two_cat_to_strict_bicat_to_two_cat (C : two_cat) : strict_bicat_to_two_cat (two_cat_to_strict_bicat C) = C. Proof. use subtypePath. { intro. do 4 (use impred ; intro). apply isapropisaset. } use total2_paths2_f. - use total2_paths2_f. + apply idpath. + apply isapropdirprod. * apply isaprop_is_precategory. intros x y. apply (pr22 (pr11 C)). * apply isaprop_has_homsets. - apply isaprop_two_cat_laws. 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.