Proposition is_double_coalg_sqr_runitor {X Y : double_coalg F} (h : double_coalg_hor X Y) : is_double_coalg_sqr (double_coalg_comp_hor_struct h (double_coalg_id_hor Y)) (sqr_of_double_coalg_hor h) (identity X) (identity Y) (runitor_h h). Proof. cbn. rewrite transportf_square_prewhisker. rewrite transportf_f_square. rewrite transportf_hcomp_r. rewrite !transportf_square_prewhisker. rewrite transportf_f_square. refine (_ @ runitor_square' _). refine (!_). etrans. { do 2 apply maponpaths. exact (lax_double_functor_runitor_h F h). } rewrite transportf_square_postwhisker. rewrite transportf_f_square. etrans. { apply maponpaths. apply square_assoc_v. } unfold transportb_square. rewrite transportf_f_square. etrans. { apply maponpaths. apply maponpaths_2. apply square_assoc_v. } unfold transportb_square. rewrite transportf_square_prewhisker. rewrite transportf_f_square. etrans. { apply maponpaths. do 2 apply maponpaths_2. refine (!_). apply comp_h_square_comp. } rewrite square_id_right_v. unfold transportb_square. rewrite transportf_hcomp_l. rewrite !transportf_square_prewhisker. rewrite transportf_f_square. use transportf_square_eq. do 2 apply maponpaths_2. rewrite transportf_hcomp_r. rewrite transportf_square_id. apply maponpaths_2. 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.