Proposition double_coalg_disp_cat_id_comp_on_twosided_disp_cat : disp_cat_id_comp E double_coalg_disp_cat_ob_mor_on_twosided_disp_cat. Proof. split. - intros X s. cbn. etrans. { do 2 apply maponpaths. apply lax_double_functor_id_square. } unfold transportb_square. rewrite transportf_square_postwhisker. rewrite transportf_f_square. rewrite square_id_right_v. unfold transportb_square. rewrite transportf_f_square. refine (_ @ !(square_id_left_v s)). unfold transportb_square. apply transportf_square_eq. apply idpath. - intros X Y Z f g h₁ h₂ h₃ s₁ s₂. cbn in s₁, s₂ ; cbn. refine (_ @ square_assoc_v' _ _ _). refine (!_). etrans. { do 2 apply maponpaths. exact (!s₂). } rewrite transportf_square_postwhisker. rewrite transportf_f_square. etrans. { apply maponpaths. apply (square_assoc_v (C := C)). } unfold transportb_square. rewrite transportf_f_square. etrans. { apply maponpaths. apply maponpaths_2. exact (!s₁). } rewrite transportf_square_prewhisker. rewrite transportf_f_square. etrans. { apply maponpaths. exact (!(square_assoc_v' _ _ _)). } rewrite transportf_f_square. refine (!_). etrans. { do 2 apply maponpaths. apply lax_double_functor_comp_v_square. } unfold transportb_square. rewrite transportf_square_postwhisker. rewrite transportf_f_square. apply 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.