Proposition transportf_hor_comp_data {C : category} {D : twosided_disp_cat C C} {x₁ x₂ y₁ y₂ z₁ z₂ : C} {vx : x₁ --> x₂} {vy : y₁ --> y₂} {vz : z₁ --> z₂} {h₁ : D x₁ y₁} {k₁ : D y₁ z₁} {h₂ : D x₂ y₂} {k₂ : D y₂ z₂} (s₁ : h₁ -->[ vx ][ vy] h₂) (s₂ : k₁ -->[ vy ][ vz] k₂) {ψ₁ ψ₂ : ∏ (x y z : C), D x y → D y z → D x z} (p : ψ₁ = ψ₂) (fg : ψ₁ x₁ y₁ z₁ h₁ k₁ -->[ vx ][ vz ] ψ₁ x₂ y₂ z₂ h₂ k₂) : transportf (λ (φ : ∏ (x y z : C), D x y → D y z → D x z), φ x₁ y₁ z₁ h₁ k₁ -->[ vx ][ vz ] φ x₂ y₂ z₂ h₂ k₂) p fg = transportf_disp_mor2 (id_right _ @ id_left _) (id_right _ @ id_left _) (idtoiso_twosided_disp (idpath _) (idpath _) (!(transportf_hor_comp_data_path p h₁ k₁)) ;;2 fg ;;2 idtoiso_twosided_disp (idpath _) (idpath _) (transportf_hor_comp_data_path p h₂ k₂)). Proof. induction p ; cbn. rewrite (id_two_disp_right (D := D)). unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite (id_two_disp_left (D := D)). unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. refine (!_). apply transportf_disp_mor2_idpath. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.StrictTwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TransportLaws. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Basics.StrictDoubleCatBasics.