Proposition lassociator_twosided_disp_nat_trans_hor_id {C₁ C₂ C₃ C₄ : category} {F : C₁ ⟶ C₂} {G : C₂ ⟶ C₃} {H : C₃ ⟶ C₄} {D₁ : twosided_disp_cat C₁ C₁} {D₂ : twosided_disp_cat C₂ C₂} {D₃ : twosided_disp_cat C₃ C₃} {D₄ : twosided_disp_cat C₄ C₄} {FF : twosided_disp_functor F F D₁ D₂} {GG : twosided_disp_functor G G D₂ D₃} {HH : twosided_disp_functor H H D₃ D₄} {I₁ : hor_id D₁} {I₂ : hor_id D₂} {I₃ : hor_id D₃} {I₄ : hor_id D₄} (FI : double_functor_hor_id FF I₁ I₂) (GI : double_functor_hor_id GG I₂ I₃) (HI : double_functor_hor_id HH I₃ I₄) : double_nat_trans_hor_id (id_twosided_disp_nat_trans (comp_twosided_disp_functor _ (comp_twosided_disp_functor _ _))) (comp_hor_id FI (comp_hor_id GI HI)) (comp_hor_id (comp_hor_id FI GI) HI). Proof. intros x ; cbn. rewrite transportf_twosided_disp_functor. rewrite !two_disp_post_whisker_f. rewrite !two_disp_pre_whisker_f. unfold transportb_disp_mor2. rewrite !transport_f_f_disp_mor2. rewrite id_two_disp_right. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite double_id_mor_id. rewrite id_two_disp_left. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite twosided_disp_functor_comp. unfold transportb_disp_mor2. rewrite !two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp. unfold transportb_disp_mor2. 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.