Lemma disp_actionbicat_disp_id_pentagon {a : B} {FA : V ⟶ category_from_bicat_and_ob a} (FAm : fmonoidal Mon_V (monoidal_from_bicat_and_ob a) FA): param_distr_bicat_pentagon_eq Mon_V FAm FAm (id₁ a) (disp_actionbicat_disp_id_nat_trans FAm). Proof. red; cbn. intros v w. unfold param_distr_bicat_pentagon_eq_body, param_distr_bicat_pentagon_eq_body_RHS. cbn. etrans. { rewrite vassocl. apply maponpaths. rewrite vassocr. apply maponpaths_2. apply vcomp_lunitor. } etrans. { repeat rewrite vassocr. apply idpath. } apply pathsinv0, (rhs_right_inv_cell _ _ _ (is_invertible_2cell_runitor _)). etrans. { rewrite !vassocl. apply idpath. } rewrite vcomp_runitor. repeat rewrite vassocr. apply maponpaths_2. rewrite <- rwhisker_vcomp. rewrite <- lwhisker_vcomp. rewrite <- runitor_triangle. rewrite <- lunitor_triangle. etrans. 2: { rewrite vassocr. rewrite rassociator_lassociator. apply pathsinv0, id2_left. } etrans. { repeat rewrite vassocr. apply maponpaths_2. repeat rewrite vassocl. rewrite lassociator_rassociator. rewrite id2_right. apply idpath. } repeat rewrite vassocl. etrans. { do 4 apply maponpaths. rewrite lwhisker_vcomp. rewrite rinvunitor_runitor. apply lwhisker_id2. } rewrite id2_right. etrans. 2: { apply id2_right. } apply maponpaths. rewrite lunitor_lwhisker. rewrite rwhisker_vcomp. rewrite rinvunitor_runitor. apply id2_rwhisker. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.Bicategories.MonoidalCategories.EndofunctorsWhiskeredMonoidal. Require Import UniMath.Bicategories.MonoidalCategories.WhiskeredMonoidalFromBicategory. Require Import UniMath.Bicategories.MonoidalCategories.ActionBasedStrongFunctorsWhiskeredMonoidal. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.Core.Invertible_2cells. Import Bicat.Notations. Import BifunctorNotations. Import DisplayedBifunctorNotations.