proposition: Lemma disp_actionbicat_disp_comp_is_nat_trans {a0 a1 a2 : B} {g1 : B ⟦ a0, a1 ⟧} {g2 : B ⟦ a1, a2 ⟧} {FA : V ⟶ category_from_bicat_and_ob a0} {FAm : fmonoidal Mon_V (monoidal_from_bicat_and_ob a0) FA} {FA' : V ⟶ category_from_bicat_and_ob a1} {FA'm : fmonoidal Mon_V (monoidal_from_bicat_and_ob a1) FA'} {FA'' : V ⟶ category_from_bicat_and_ob a2} {FA''m : fmonoidal Mon_V (monoidal_from_bicat_and_ob a2) FA''} (Hyp1 : disp_actionbicat_disp_mor FAm FA'm g1) (Hyp2 : disp_actionbicat_disp_mor FA'm FA''m g2): is_nat_trans _ _ (disp_actionbicat_disp_comp_nat_trans_data Hyp1 Hyp2).
proof: Proof. intros v w f. unfold disp_actionbicat_disp_comp_nat_trans_data. cbn; rewrite vassocr. rewrite (! lwhisker_lwhisker_rassociator _ _ _ _ _ _ _ _ _). rewrite vassocr. etrans. { apply maponpaths_2. rewrite vassocl. apply maponpaths. exact (lwhisker_vcomp g1 (g2 ◃ # FA'' f) (pr1 Hyp2 w)). } etrans. 2: { rewrite !vassocr. rewrite vassocl. apply maponpaths. apply rwhisker_rwhisker_alt. } rewrite !vassocl. apply maponpaths. rewrite !vassocr. apply maponpaths_2. etrans. { apply maponpaths_2. apply maponpaths_2. apply maponpaths. exact (pr21 Hyp2 v w f). } etrans. 2: { rewrite vassocl. apply maponpaths. rewrite rwhisker_vcomp. apply maponpaths. exact (pr21 Hyp1 v w f). } cbn. rewrite (! lwhisker_vcomp _ _ _). do 3 rewrite vassocl. apply maponpaths. rewrite vassocr. etrans. 2: { rewrite (! rwhisker_vcomp _ _ _). rewrite vassocr. apply idpath. } apply maponpaths_2. apply rwhisker_lwhisker. Qed.
imports: UniMath.Foundations.PartD UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.Bicategories.MonoidalCategories.EndofunctorsWhiskeredMonoidal UniMath.Bicategories.MonoidalCategories.WhiskeredMonoidalFromBicategory UniMath.Bicategories.MonoidalCategories.ActionBasedStrongFunctorsWhiskeredMonoidal UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.Unitors UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.Core.Invertible_2cells