Proposition pre_whisker_twosided_disp_nat_trans_hor_id {C₁ C₂ C₃ : category} {F : C₁ ⟶ C₂} {G G' : C₂ ⟶ C₃} {τ : G ⟹ G'} {D₁ : twosided_disp_cat C₁ C₁} {D₂ : twosided_disp_cat C₂ C₂} {D₃ : twosided_disp_cat C₃ C₃} {FF : twosided_disp_functor F F D₁ D₂} {GG : twosided_disp_functor G G D₂ D₃} {GG' : twosided_disp_functor G' G' D₂ D₃} {ττ : twosided_disp_nat_trans τ τ GG GG'} {I₁ : hor_id D₁} {I₂ : hor_id D₂} {I₃ : hor_id D₃} (FFI : double_functor_hor_id FF I₁ I₂) {GGI : double_functor_hor_id GG I₂ I₃} {GGI' : double_functor_hor_id GG' I₂ I₃} (ττI : double_nat_trans_hor_id ττ GGI GGI') : double_nat_trans_hor_id (pre_whisker_twosided_disp_nat_trans FF ττ) (comp_hor_id FFI GGI) (comp_hor_id FFI GGI'). Proof. intros x ; cbn. etrans. { rewrite two_disp_pre_whisker_f. rewrite assoc_two_disp_alt. rewrite transport_f_f_disp_mor2. rewrite (pr2 ττ). rewrite two_disp_post_whisker_b. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. etrans. { apply maponpaths. apply maponpaths_2. exact (ττI (F x)). } rewrite two_disp_pre_whisker_b. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. apply idpath. } refine (!_). etrans. { rewrite two_disp_post_whisker_f. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. apply idpath. } 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.