Proposition double_cat_weq_univalent_doublecategory_inv₂ (C : univalent_doublecategory) : double_cat_to_univalent_doublecategory (doublecategory_to_univalent_double_cat C) = C. Proof. use subtypePath. { intro. apply isapropdirprod. - apply isaprop_is_univalent. - apply isaprop_is_univalent_twosided_disp_cat. } use subtypePath. { intro. use isapropdirprod ; repeat (use impred ; intro) ; apply isapropiscontr. } use total2_paths_f ; [ apply idpath | ]. repeat (use pathsdirprod) ; repeat (use funextsec ; intro) ; apply get_has_sq_hor_homsets. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.BicatOfDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.AlternativeDefinitions.DoubleCatsUnfolded.