Proposition double_coalg_id_is_sqr {X Y : double_coalg F} (v : double_coalg_ver X Y) : is_double_coalg_sqr (double_coalg_id_hor_struct X) (double_coalg_id_hor_struct Y) _ _ (id_h_square v). Proof. cbn. rewrite transportf_square_postwhisker. rewrite transportf_square_prewhisker. rewrite transportf_f_square. etrans. { apply maponpaths. refine (!_). apply (square_assoc_v' (C := C)). } rewrite transportf_f_square. etrans. { do 2 apply maponpaths. apply (lax_double_functor_id_h_mor' F). } rewrite transportf_square_postwhisker. rewrite transportf_f_square. etrans. { apply maponpaths. apply (square_assoc_v (C := C)). } unfold transportb_square. rewrite transportf_f_square. rewrite <- id_h_square_comp. etrans. { apply maponpaths. apply maponpaths_2. exact (id_h_square_eq (coalgebra_mor_commutes F v)). } rewrite transportf_square_prewhisker. rewrite transportf_f_square. refine (!_). etrans. { apply maponpaths. apply (square_assoc_v (C := C)). } unfold transportb_square. rewrite transportf_f_square. rewrite <- id_h_square_comp. use transportf_square_eq. apply idpath. Qed.
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.