Lemma bracket_Thm15_ok_part1 (f : Ptd ⟦ Z, ptd_from_alg InitAlg ⟧) : # U f = # (pr1 (ℓ (U Z))) (η InitAlg) · ⦃f⦄. Proof. apply nat_trans_eq_alt; intro c. assert (h_eq := pr2 (pr1 (SpecializedGMIt_Thm15 f))). assert (h_eq' := maponpaths (λ m, ((aux_iso_1_inv:(_⟹_)) _)· m) h_eq); clear h_eq. assert (h_eq1' := maponpaths (λ m, (BinCoproductIn1 (CPEndC _ _))· m) h_eq'); clear h_eq'. assert (h_eq1'_inst := nat_trans_eq_pointwise h_eq1' c); clear h_eq1'. eapply pathscomp0, pathscomp0; [|apply (!h_eq1'_inst)|]; clear h_eq1'_inst. - apply BinCoproductIn1Commutes_right_in_ctx_dir; simpl. rewrite id_left, <- !assoc. apply BinCoproductIn1Commutes_right_in_ctx_dir; simpl. rewrite !id_left, !(@id_left EndC). now apply BinCoproductIn1Commutes_right_in_ctx_dir, BinCoproductIn1Commutes_right_in_ctx_dir, BinCoproductIn1Commutes_right_dir. - apply BinCoproductIn1Commutes_left_in_ctx_dir; simpl. now rewrite id_left, assoc. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. 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.FunctorAlgebras. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Require Import UniMath.CategoryTheory.Chains.All. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.Categories.HSET.Core. 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.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration_alt. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Notation.