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 : category_Ptd C ⟦ Z, ptd_from_alg (InitAlg C CP H IA)⟧), # U 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.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.SubstitutionSystems.LiftingInitial. Require Import UniMath.SubstitutionSystems.MonadsFromSubstitutionSystems. Require Import UniMath.SubstitutionSystems.LamSignature. Require Import UniMath.SubstitutionSystems.Lam. Require Import UniMath.SubstitutionSystems.Notation.