Lemma ptdlaxcommutator_genopt_tensor : relativelaxcommutator_tensor Mon_endo_CAT Mon_ptdendo_CAT (forget_monoidal_pointed_objects_monoidal Mon_endo_CAT) ptdlaxcommutator_genopt_data. Proof. intros Ze Z'e'; induction Ze as [Z e]; induction Z'e' as [Z' e']. apply nat_trans_eq_alt; intro c; simpl. unfold δ_genoption_mor, BinCoproduct_of_functors_ob, BinCoproduct_of_functors_mor; simpl. repeat rewrite id_right. rewrite id_left. rewrite precompWithBinCoproductArrow. rewrite postcompWithBinCoproductArrow. apply maponpaths_12. - rewrite id_left. cbn in Z, e, Z', e'. etrans. 2: { rewrite assoc'. apply maponpaths. apply (nat_trans_ax e'). } simpl. do 2 rewrite assoc. rewrite BinCoproductIn1Commutes. apply idpath. - rewrite id_left. etrans. 2: { apply functor_comp. } apply pathsinv0, maponpaths, BinCoproductIn2Commutes. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Combinatorics.Lists. 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.whiskering. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.coslicecat. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Limits. Require Import UniMath.CategoryTheory.Categories.HSET.Colimits. Require Import UniMath.CategoryTheory.Categories.HSET.Structures. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Actegories.Actegories. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories. Require Import UniMath.CategoryTheory.Actegories.MorphismsOfActegories. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegoryMorphisms. Require Import UniMath.CategoryTheory.Actegories.CoproductsInActegories. Require Import UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects. Require Import UniMath.CategoryTheory.Monoidal.Examples.EndofunctorsMonoidalElementary. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonadsAsMonoidsElementary. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.BindingSigToMonad. Require Import UniMath.SubstitutionSystems.GeneralizedSubstitutionSystems. Require Import UniMath.SubstitutionSystems.ConstructionOfGHSS. Require Import UniMath.SubstitutionSystems.SigmaMonoids. Require Import UniMath.SubstitutionSystems.Notation. Import MonoidalNotations.