proposition: Lemma strength_law2_pw : ∏ (X : EndC) (Z Z' : Ptd) x, ((theta H) (X ⊗ (Z p• Z')) : nat_trans _ _) x = ((theta H) (X ⊗ Z') •• (U Z):nat_trans _ _) x · ((theta H) ((functor_compose (U Z') X) ⊗ Z):nat_trans _ _) x · (# H (identity (functor_compose (U Z ∙ U Z') X) : [C, C] ⟦ functor_compose (U Z) (U Z' ∙ X : [C, C]), functor_compose (U Z ∙ U Z') X ⟧) : nat_trans _ _) x.
proof: Proof. intros X Z Z' x. etrans; revgoals. { apply (nat_trans_eq_pointwise (θ_Strength2_int_implies_θ_Strength2 _ (Sig_strength_law2 H) X Z Z' _ (identity _) ) x). } etrans;[eapply pathsinv0;apply id_right|]. apply cancel_precomposition. apply pathsinv0. etrans. { eapply nat_trans_eq_pointwise. apply (functor_id H). } apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.MoreFoundations.Propositions UniMath.CategoryTheory.PointedFunctors UniMath.CategoryTheory.PointedFunctorsComposition UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.FunctorCategory UniMath.SubstitutionSystems.Signatures UniMath.SubstitutionSystems.Notation UniMath.CategoryTheory.Monads.LModules UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems UniMath.CategoryTheory.Monads.Monads UniMath.CategoryTheory.whiskering UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems UniMath.CategoryTheory.HorizontalComposition UniMath.CategoryTheory.FunctorAlgebras UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Chains.All UniMath.CategoryTheory.Presheaf UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt UniMath.SubstitutionSystems.GenMendlerIteration_alt UniMath.CategoryTheory.Limits.Initial