proposition: Proposition pre_whisker_twosided_disp_nat_trans_hor_comp {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'} {Cm₁ : hor_comp D₁} {Cm₂ : hor_comp D₂} {Cm₃ : hor_comp D₃} (FFC : double_functor_hor_comp FF Cm₁ Cm₂) {GGC : double_functor_hor_comp GG Cm₂ Cm₃} {GGC' : double_functor_hor_comp GG' Cm₂ Cm₃} (ττC : double_nat_trans_hor_comp ττ GGC GGC') : double_nat_trans_hor_comp (pre_whisker_twosided_disp_nat_trans FF ττ) (comp_hor_comp FFC GGC) (comp_hor_comp FFC GGC').
proof: Proof. intros x y z h k ; cbn. rewrite two_disp_pre_whisker_b. rewrite two_disp_post_whisker_b. rewrite transport_b_b_disp_mor2. rewrite assoc_two_disp_alt. unfold transportb_disp_mor2. 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. rewrite ττC. 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. use transportf_disp_mor2_eq. apply idpath. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor