Definition double_coalg_twosided_iso_weq {X Y : double_coalg F} (h k : double_coalg_hor X Y) : (∑ (s : globular_iso_square h k), is_double_coalg_sqr (sqr_of_double_coalg_hor h) (sqr_of_double_coalg_hor k) (identity X) (identity Y) s) ≃ iso_twosided_disp (identity_z_iso X) (identity_z_iso Y) h k. Proof. use weq_iso. - exact (λ s, to_double_coalg_twosided_iso (pr1 s) (pr2 s)). - exact from_double_coalg_twosided_iso. - abstract (intro s ; use subtypePath ; [ intro ; apply isaset_square | ] ; use subtypePath ; [ intro ; apply isaprop_is_iso_twosided_disp | ] ; apply idpath). - abstract (intro s ; use subtypePath ; [ intro ; apply isaprop_is_iso_twosided_disp | ] ; use eq_double_coalg_sqr ; apply idpath). Defined.
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.FunctorCoalgebras. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Total. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.DispCatOnTwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.TwoSidedDispCatOnDispCat. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.DerivedLaws. Import TransportSquare.