Proposition double_cat_weq_univalent_doublecategory_inv₁ (C : univalent_double_cat) : doublecategory_to_univalent_double_cat (double_cat_to_univalent_doublecategory C) = C. Proof. induction C as [ C1 H ]. use subtypePath. { intro. apply isapropdirprod ; repeat (use impred ; intro) ; apply isaset_disp_mor. } use total2_paths_f ; [ apply idpath | ]. use pathsdirprod. - use subtypePath. { intro. apply isaprop_double_lunitor_laws. } apply idpath. - use pathsdirprod. + use subtypePath. { intro. apply isaprop_double_runitor_laws. } apply idpath. + use subtypePath. { intro. apply isaprop_double_associator_laws. } apply idpath. 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.