proposition: Lemma bracket_Thm15_ok_part2 (f : [C, C] ⟦ U Z, U (ptd_from_alg InitAlg) ⟧) : prestrength_in_first_arg (alg_carrier _ InitAlg) · # H ⦃f⦄ · τ InitAlg = # (pr1 (ℓ (U Z))) (τ InitAlg) · ⦃f⦄.
proof: 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_eq2' := maponpaths (λ m, (BinCoproductIn2 (CPEndC _ _))· m) h_eq'); clear h_eq'. assert (h_eq2'_inst := nat_trans_eq_pointwise h_eq2' c); clear h_eq2'. eapply pathscomp0, pathscomp0; [|apply (!h_eq2'_inst)|]; clear h_eq2'_inst. - apply BinCoproductIn2Commutes_right_in_ctx_dir; simpl. rewrite id_right, id_left, <- !assoc. apply BinCoproductIn2Commutes_right_in_ctx_dir; simpl. rewrite id_left. apply BinCoproductIn2Commutes_right_in_ctx_dir. apply BinCoproductIn2Commutes_right_in_double_ctx_dir. rewrite <- assoc; apply maponpaths. apply pathsinv0; simpl. rewrite <- assoc; apply maponpaths. now apply BinCoproductIn2Commutes. - apply BinCoproductIn2Commutes_left_in_ctx_dir. now rewrite id_left; apply assoc. 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.Limits.Graphs.Colimits UniMath.CategoryTheory.Chains.All 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.SimplifiedHSS.SubstitutionSystems UniMath.SubstitutionSystems.GenMendlerIteration_alt UniMath.CategoryTheory.BicatOfCatsElementary UniMath.SubstitutionSystems.Notation