proposition: Proposition lassociator_twosided_disp_nat_trans_hor_comp {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} {Cm₁ : hor_comp D₁} {Cm₂ : hor_comp D₂} {Cm₃ : hor_comp D₃} {Cm₄ : hor_comp D₄} (FC : double_functor_hor_comp FF Cm₁ Cm₂) (GC : double_functor_hor_comp GG Cm₂ Cm₃) (HC : double_functor_hor_comp HH Cm₃ Cm₄) : double_nat_trans_hor_comp (id_twosided_disp_nat_trans (comp_twosided_disp_functor _ (comp_twosided_disp_functor _ _))) (comp_hor_comp FC (comp_hor_comp GC HC)) (comp_hor_comp (comp_hor_comp FC GC) HC).
proof: Proof. intros x y z h k ; cbn. rewrite transportb_twosided_disp_functor. rewrite two_disp_post_whisker_b. rewrite transport_b_b_disp_mor2. rewrite id_two_disp_right. rewrite transport_b_b_disp_mor2. rewrite two_disp_post_whisker_b. rewrite two_disp_pre_whisker_b. rewrite two_disp_post_whisker_b. rewrite !transport_b_b_disp_mor2. rewrite double_hor_comp_mor_id. rewrite id_two_disp_left. rewrite transport_b_b_disp_mor2. rewrite twosided_disp_functor_comp. rewrite two_disp_post_whisker_b. rewrite transport_b_b_disp_mor2. rewrite assoc_two_disp. rewrite transport_b_b_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