Proposition comp_twosided_disp_nat_trans_hor_comp {C₁ C₂ : category} {F F' F'' : C₁ ⟶ C₂} {τ : F ⟹ F'} {τ' : F' ⟹ F''} {D : twosided_disp_cat C₁ C₁} {D' : twosided_disp_cat C₂ C₂} {FF : twosided_disp_functor F F D D'} {FF' : twosided_disp_functor F' F' D D'} {FF'' : twosided_disp_functor F'' F'' D D'} {ττ : twosided_disp_nat_trans τ τ FF FF'} {ττ' : twosided_disp_nat_trans τ' τ' FF' FF''} {Cm : hor_comp D} {Cm' : hor_comp D'} {FFC : double_functor_hor_comp FF Cm Cm'} {FFC' : double_functor_hor_comp FF' Cm Cm'} {FFC'' : double_functor_hor_comp FF'' Cm Cm'} (ττC : double_nat_trans_hor_comp ττ FFC FFC') (ττC' : double_nat_trans_hor_comp ττ' FFC' FFC'') : double_nat_trans_hor_comp (comp_twosided_disp_nat_trans ττ ττ') FFC FFC''. Proof. intros x y z h k ; cbn. rewrite assoc_two_disp. rewrite (ττC x). rewrite two_disp_pre_whisker_b. rewrite transport_b_b_disp_mor2. rewrite assoc_two_disp_alt. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. unfold transportb. rewrite (ττC' x). rewrite two_disp_post_whisker_b. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite double_hor_comp_mor_comp. rewrite assoc_two_disp_alt. rewrite transport_f_f_disp_mor2. use transportf_disp_mor2_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.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor.