proposition: Proposition comp_functor_associator {C₁ C₂ C₃ : category} {D₁ : twosided_disp_cat C₁ C₁} {D₂ : twosided_disp_cat C₂ C₂} {D₃ : twosided_disp_cat C₃ C₃} {Cm₁ : hor_comp D₁} {Cm₂ : hor_comp D₂} {Cm₃ : hor_comp D₃} {a₁ : double_cat_associator Cm₁} {a₂ : double_cat_associator Cm₂} {a₃ : double_cat_associator Cm₃} {F : C₁ ⟶ C₂} {FF : twosided_disp_functor F F D₁ D₂} {G : C₂ ⟶ C₃} {GG : twosided_disp_functor G G D₂ D₃} {FC : double_functor_hor_comp FF Cm₁ Cm₂} {GC : double_functor_hor_comp GG Cm₂ Cm₃} (Fa : double_functor_associator a₁ a₂ FC) (Ga : double_functor_associator a₂ a₃ GC) : double_functor_associator a₁ a₃ (comp_hor_comp FC GC).
proof: Proof. intros w x y z f g h ; cbn. rewrite !two_disp_post_whisker_b. rewrite two_disp_pre_whisker_b. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. etrans. { etrans. { apply maponpaths. apply maponpaths_2. apply maponpaths. apply maponpaths. exact (id_two_disp_left_alt _). } rewrite (double_hor_comp_transport_mor _ _ _ _ _ _ _ (functor_id _ _)). rewrite two_disp_post_whisker_f. rewrite two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. etrans. { do 3 apply maponpaths. rewrite assoc_two_disp. apply maponpaths. apply maponpaths_2. rewrite twosided_disp_functor_id_alt. rewrite functor_double_comp_eq_transport. apply idpath. } unfold transportb_disp_mor2. rewrite !two_disp_post_whisker_f. rewrite two_disp_pre_whisker_f. 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 assoc_two_disp. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp. unfold transportb_disp_mor2. rewrite two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. etrans. { apply maponpaths. do 2 apply maponpaths_2. apply Ga. } rewrite !two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. 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 twosided_disp_functor_comp_alt. rewrite !two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. etrans. { do 3 apply maponpaths. rewrite assoc_two_disp. rewrite Fa. unfold transportb_disp_mor2. rewrite transport_f_f_disp_mor2. rewrite transportf_twosided_disp_functor. apply idpath. } rewrite !two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. do 2 rewrite twosided_disp_functor_comp. unfold transportb_disp_mor2. rewrite two_disp_pre_whisker_f. rewrite !two_disp_post_whisker_f. rewrite !transport_f_f_disp_mor2. apply idpath. } refine (!_). etrans. { etrans. { apply maponpaths. do 2 apply maponpaths_2. apply maponpaths_2. exact (id_two_disp_left_alt _). } rewrite (double_hor_comp_transport_mor _ _ _ _ _ _ _ (!(functor_id _ _))). rewrite !two_disp_pre_whisker_f. rewrite !transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite !transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite !transport_f_f_disp_mor2. etrans. { do 2 apply maponpaths. rewrite assoc_two_disp. apply maponpaths. apply maponpaths_2. rewrite assoc_two_disp. apply maponpaths. apply maponpaths_2. rewrite twosided_disp_functor_id_alt. rewrite transport_f_f_disp_mor2. apply (functor_double_comp_eq_transport GC _ _ (idpath _) (idpath _)). } unfold transportb_disp_mor2. rewrite !two_disp_post_whisker_f. rewrite !two_disp_pre_whisker_f. rewrite !two_disp_post_whisker_f. rewrite !transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite !two_disp_post_whisker_f. rewrite !transport_f_f_disp_mor2. rewrite assoc_two_disp_alt. rewrite !two_disp_post_whisker_f. rewrite !transport_f_f_disp_mor2. apply idpath. } rewrite assoc_two_disp_alt. rewrite !two_disp_post_whisker_f. rewrite !transport_f_f_disp_mor2. use transportf_disp_mor2_eq. apply idpath. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics UniMath.Bicategories.DoubleCategories.Core.DoubleFunctor.Basics