Lemma bracket_Thm15_ok_part2 (Z: Ptd)(f : [C, C] ⟦ U Z, U (ptd_from_alg InitAlg) ⟧): (theta H) ((alg_carrier _ InitAlg) ⊗ Z) · # H ⦃f⦄ · τ InitAlg = # (pr1 (ℓ (U Z))) (τ InitAlg) · ⦃f⦄. Proof. apply nat_trans_eq_alt. intro c. assert (h_eq := pr2 (pr1 (SpecializedGMIt_Thm15 Z f))). assert (h_eq' := maponpaths (fun m:EndC⟦_,pr1 InitAlg⟧ => (((aux_iso_1_inv Z):(_⟹_)) _)· m) h_eq); clear h_eq. *) assert (h_eq2' := maponpaths (fun m:EndC⟦_,pr1 InitAlg⟧ => (BinCoproductIn2 (CPEndC _ _))· m) h_eq'). clear h_eq'. assert (h_eq2'_inst := nat_trans_eq_pointwise h_eq2' c). clear h_eq2'. match goal with |[ H1 : _ = ?f |- _ = _ ] => intermediate_path f end. - clear h_eq2'_inst. apply BinCoproductIn2Commutes_right_in_ctx_dir. unfold aux_iso_1; simpl. rewrite id_right. rewrite id_left. do 3 rewrite <- assoc. apply BinCoproductIn2Commutes_right_in_ctx_dir. unfold nat_trans_id; simpl. rewrite id_left. apply BinCoproductIn2Commutes_right_in_ctx_dir. unfold nat_trans_fix_snd_arg_data. apply BinCoproductIn2Commutes_right_in_double_ctx_dir. rewrite <- assoc. apply maponpaths. eapply pathscomp0. 2: apply assoc. apply maponpaths. apply pathsinv0. apply BinCoproductIn2Commutes. - rewrite <- h_eq2'_inst. clear h_eq2'_inst. apply BinCoproductIn2Commutes_left_in_ctx_dir. repeat rewrite id_left. apply 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.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.SimplifiedHSS.SubstitutionSystems. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.RightKanExtension. Require Import UniMath.SubstitutionSystems.GenMendlerIteration. Require Import UniMath.CategoryTheory.BicatOfCatsElementary. Require Import UniMath.SubstitutionSystems.Notation.