proposition: Lemma dialgebra_disp_leftunitor_data : disp_leftunitor_data dialgebra_disp_tensor dialgebra_disp_unit.
proof: Proof. intros a f. cbn. unfold dialgebra_disp_unit, dialgebra_disp_tensor_op. rewrite (bifunctor_equalwhiskers W). unfold functoronmorphisms2. rewrite bifunctor_rightcomp. set (aux1 := fmonoidal_preservesleftunitality Gm a). etrans. { repeat rewrite assoc'. do 3 apply maponpaths. rewrite assoc. exact aux1. } clear aux1. apply (z_iso_inv_on_right _ _ _ (_,,fmonoidal_preservestensorstrongly Fm I_{V} a)). cbn. set (aux2 := fmonoidal_preservesleftunitality Fm a). etrans. { rewrite assoc. apply cancel_postcomposition. apply pathsinv0, (bifunctor_equalwhiskers W). } unfold functoronmorphisms1. rewrite assoc'. etrans. { apply maponpaths. apply monoidal_leftunitornat. } repeat rewrite assoc. apply cancel_postcomposition. rewrite <- aux2. clear aux2. repeat rewrite assoc. apply cancel_postcomposition. rewrite <- (bifunctor_rightcomp W). etrans. { apply cancel_postcomposition. apply maponpaths. apply (z_iso_after_z_iso_inv (_,,fmonoidal_preservesunitstrongly Fm)). } rewrite bifunctor_rightid. apply id_left. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Categories.Dialgebras UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Displayed.MonoidalSections UniMath.CategoryTheory.Monoidal.Displayed.Symmetric UniMath.CategoryTheory.Monoidal.Displayed.MonoidalFunctorLifting UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.Monoidal.Structure.Symmetric