proposition: Proposition post_whisker_twosided_disp_nat_trans_hor_comp {C₁ C₂ C₃ : category} {F F' : C₁ ⟶ C₂} {G : C₂ ⟶ C₃} {τ : F ⟹ F'} {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} {FF' : twosided_disp_functor F' F' D₁ D₂} {GG : twosided_disp_functor G G D₂ D₃} {ττ : twosided_disp_nat_trans τ τ FF FF'} {Cm₁ : hor_comp D₁} {Cm₂ : hor_comp D₂} {Cm₃ : hor_comp D₃} {FFC : double_functor_hor_comp FF Cm₁ Cm₂} {FFC' : double_functor_hor_comp FF' Cm₁ Cm₂} (GGC : double_functor_hor_comp GG Cm₂ Cm₃) (ττC : double_nat_trans_hor_comp ττ FFC FFC') : double_nat_trans_hor_comp (post_whisker_twosided_disp_nat_trans GG 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 assoc_two_disp_alt. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite twosided_disp_functor_comp_alt. rewrite two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. rewrite (ττC x y z h k). rewrite transportb_twosided_disp_functor. rewrite two_disp_post_whisker_b. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite twosided_disp_functor_comp. 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 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. rewrite (functor_double_comp_eq GGC _ _). unfold transportb_disp_mor2. rewrite two_disp_pre_whisker_f. 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