Lemma bracket_Thm15_ok_η : ∏ (C : category) (CP : BinCoproducts C) (KanExt : ∏ Z : category_Ptd C, GlobalRightKanExtensionExists C C (U Z) C) (H : Presignature C C C) (IA : Initial (FunctorAlg (Id_H C CP H))) (Z : category_Ptd C) (f : [C,C] ⟦ U Z, U (ptd_from_alg (InitAlg C CP H IA))⟧), f = # (pr1 (ℓ (U Z))) (η (InitAlg C CP H IA)) · bracket_Thm15 C CP KanExt H IA Z f. Proof. apply bracket_Thm15_ok_part1. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.FunctorAlgebras. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.PointedFunctors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.HorizontalComposition. Require Import UniMath.CategoryTheory.PointedFunctorsComposition. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.BinSumOfSignatures. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.LiftingInitial. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.SimplifiedHSS.Lam. Require Import UniMath.SubstitutionSystems.Notation.