Lemma mhss_second_monoidlaw : monoid_laws_unit_left Mon_V mhss_monoid_data. Proof. red. cbn. etrans. 2: { apply μ_1_is_instance_of_left_unitor. } apply (mfbracket_unique(Z:=I_{Mon_PtdV})). split. - exact mhss_second_monoidlaw_aux. - rewrite functor_comp. transitivity (μ_0 ⊗^{ Mon_V}_{r} H (pr1 gh) · θ Ptd_from_mhss (pr1 gh) · # H μ · τ). { apply cancel_postcomposition. rewrite assoc. apply cancel_postcomposition. apply pathsinv0. set (aux := lineator_linnatright Mon_PtdV (actegory_with_canonical_pointed_action Mon_V) (actegory_with_canonical_pointed_action Mon_V) H θ I_{ Mon_PtdV} Ptd_from_mhss (pr1 gh) μ_0_Ptd). cbn in aux. etrans. { exact aux. } apply idpath. } etrans. { do 2 rewrite assoc'. apply maponpaths. rewrite assoc. apply (mfbracket_τ(Z:=Ptd_from_mhss)). } repeat rewrite assoc. apply cancel_postcomposition. cbn. apply (bifunctor_equalwhiskers Mon_V). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids. Require Import UniMath.CategoryTheory.Actegories.Actegories. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories. Require Import UniMath.CategoryTheory.Actegories.MorphismsOfActegories. Require Import UniMath.CategoryTheory.Actegories.CoproductsInActegories. Require Import UniMath.CategoryTheory.coslicecat. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects. Import BifunctorNotations. Import MonoidalNotations.