Definition from_double_coalg_twosided_iso {X Y : double_coalg F} {h k : double_coalg_hor X Y} (s : iso_twosided_disp (identity_z_iso X) (identity_z_iso Y) h k) : ∑ (s : globular_iso_square h k), is_double_coalg_sqr (sqr_of_double_coalg_hor h) (sqr_of_double_coalg_hor k) (identity _) (identity _) s. Proof. simple refine ((_ ,, _ ,, _ ,, _) ,, _). - exact (pr11 s). - exact (pr1 (pr12 s)). - abstract (refine (maponpaths pr1 (pr122 s) @ _) ; refine (transportb_double_coalg_sqr (id_left _) (id_left _) _ @ _) ; use transportf_square_eq ; apply idpath). - abstract (refine (maponpaths pr1 (pr222 s) @ _) ; refine (transportb_double_coalg_sqr (id_left _) (id_left _) _ @ _) ; use transportf_square_eq ; apply idpath). - exact (pr21 s). 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.