Proposition transpose_triangle_law : triangle_law transpose_lunitor transpose_runitor transpose_associator. Proof. unfold triangle_law. unfold double_associator, double_lunitor, double_runitor. cbn -[idtoiso_twosided_disp]. intros x y z v w. etrans. { apply maponpaths_2. apply idtoiso_twosided_disp_transpose. } etrans. { apply maponpaths. apply maponpaths_2. apply idtoiso_twosided_disp_transpose. } refine (!_). etrans. { do 2 apply maponpaths. apply idtoiso_twosided_disp_transpose. } rewrite transportb_disp_mor2_transpose. etrans. { do 2 apply maponpaths. refine (!_). apply path_to_globular_iso_square_inv. } rewrite !globular_iso_to_path_to_iso. rewrite transportf_square_prewhisker. rewrite !transportf_square_postwhisker. rewrite transportf_square_prewhisker. rewrite transportf_f_square. etrans. { apply maponpaths. apply maponpaths_2. refine (!_). apply (lunitor_square' (C := C)). } rewrite !transportf_square_prewhisker. rewrite transportf_f_square. rewrite <- square_assoc_v'. rewrite transportf_f_square. etrans. { do 2 apply maponpaths. apply lunitor_linvunitor_h. } unfold transportb_square. rewrite transportf_square_postwhisker. rewrite !transportf_f_square. rewrite square_id_right_v. unfold transportb_square. rewrite transportf_f_square. rewrite !id_h_square_comp. rewrite !id_h_square_id. rewrite square_id_right_v. rewrite square_id_left_v. unfold transportb_square. rewrite !transportf_square_prewhisker. rewrite !transportf_square_postwhisker. rewrite !transportf_f_square. rewrite !transportf_hcomp_r. rewrite !transportf_f_square. use transportf_square_eq. apply maponpaths_2. use transportf_square_eq. apply idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Comma. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.DerivedLaws. Import TransportSquare.