Lemma j_mon_η : ∏ a : C, (Monads.η T_mon) a · j_mor a = (Monads.η M) a. Proof. intro a. etrans; [ apply assoc' |]. etrans. { apply cancel_precomposition. apply (nat_trans_eq_pointwise (algebra_mor_commutes _ _ _ j) a). } etrans; [ apply assoc |]. etrans. { apply cancel_postcomposition. apply BinCoproductIn1Commutes. } etrans; [ apply assoc' |]. etrans. { apply cancel_precomposition. apply BinCoproductIn1Commutes. } apply id_left. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.Notation. Require Import UniMath.CategoryTheory.Monads.LModules. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.Limits.Initial.