Definition SignatureToHSSisInitial (IC : Initial C) (CLC : Colims_of_shape nat_graph C) (H : Presignature C C C) (Hs : is_omega_cocont H) : isInitial _ (SignatureToHSS IC CLC H Hs). Proof. now unfold SignatureToHSS; destruct InitialHSS. Qed. Let Monad_from_hss (H : Signature C C C) : HSS H → Monad C. Proof. exact (Monad_from_hss _ BCC H). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Combinatorics.Lists. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. 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.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.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.SignatureExamples. Require Import UniMath.SubstitutionSystems.SumOfSignatures. Require Import UniMath.SubstitutionSystems.BinProductOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial_alt. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.Notation.