Proposition transportb_disp_mor2_transpose {x₁ x₂ y₁ y₂ : transpose_category} {h h' : x₁ --> x₂} {k k' : y₁ --> y₂} {v₁ : transpose_twosided_disp_cat_data x₁ y₁} {v₂ : transpose_twosided_disp_cat_data x₂ y₂} (p : h = h') (q : k = k') (s : v₁ -->[ h' ][ k' ] v₂) : transportb_disp_mor2 p q s = transportf_square (id_v_right _ @ id_v_left _) (id_v_right _ @ id_v_left _) (path_to_globular_iso_square p ⋆v s ⋆v path_to_globular_iso_square (!q)). Proof. induction p, q ; cbn. cbn in h, k. rewrite <- !(path_to_globular_iso_square_id (C := C)). rewrite square_id_right_v. rewrite square_id_left_v. unfold transportb_square. rewrite !transportf_f_square. refine (!_). apply transportf_square_id. 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.