proposition: Lemma j_mon_square_eq1 : # L (alg_map Id_H T_alg) · ((μ T_mon : EndC ⟦_, _⟧) · j_mor) = (ψ : nat_trans _ _) `T_alg ((μ T_mon : EndC ⟦_, _⟧) · j_mor).
proof: Proof. apply coprod_iter_eq; intro x. - etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. apply (Monad_law1 (T:=T_mon)). } apply id_left. - etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. apply (LModule_Mor_σ _ τT). } etrans; [ apply assoc' |]. etrans; [ apply assoc' |]. etrans; [| apply assoc ]. apply cancel_precomposition. rewrite functor_comp. etrans; [| apply assoc ]. apply cancel_precomposition. apply j_mor_rep. 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