proposition: Lemma bracket_Thm15_ok_part1 (Z: Ptd)(f : Ptd ⟦ Z, ptd_from_alg InitAlg ⟧): # U f = # (pr1 (ℓ (U Z))) (η InitAlg) · ⦃f⦄.
proof: 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. simpl in h_eq'. assert (h_eq1' := maponpaths (fun m:EndC⟦_,pr1 InitAlg⟧ => (BinCoproductIn1 (CPEndC _ _))· m) h_eq'); clear h_eq'. assert (h_eq1'_inst := nat_trans_eq_pointwise h_eq1' c); clear h_eq1'. match goal with |[ H1 : _ = ?f |- _ = _ ] => intermediate_path f end. - clear h_eq1'_inst. unfold coproduct_nat_trans_data; simpl. unfold coproduct_nat_trans_in1_data ; simpl. repeat rewrite <- assoc . apply BinCoproductIn1Commutes_right_in_ctx_dir. simpl. rewrite id_left. apply BinCoproductIn1Commutes_right_in_ctx_dir. simpl. rewrite id_left. apply BinCoproductIn1Commutes_right_in_ctx_dir. rewrite (@id_left EndC). rewrite id_left. apply BinCoproductIn1Commutes_right_in_ctx_dir. rewrite (@id_left EndC). apply BinCoproductIn1Commutes_right_dir. apply idpath. - rewrite <- h_eq1'_inst. clear h_eq1'_inst. apply BinCoproductIn1Commutes_left_in_ctx_dir. unfold nat_trans_id; simpl. rewrite id_left. repeat rewrite (id_left EndEndC). repeat rewrite (id_left EndC). unfold functor_fix_snd_arg_ob. repeat rewrite assoc. apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.MoreFoundations.PartA UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Monads.Monads UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.FunctorAlgebras UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.yoneda UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.PointedFunctors UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.HorizontalComposition UniMath.CategoryTheory.PointedFunctorsComposition UniMath.SubstitutionSystems.Signatures UniMath.SubstitutionSystems.SubstitutionSystems UniMath.SubstitutionSystems.GenMendlerIteration UniMath.CategoryTheory.RightKanExtension UniMath.SubstitutionSystems.GenMendlerIteration UniMath.CategoryTheory.BicatOfCatsElementary UniMath.SubstitutionSystems.Notation