proposition: Proposition functor_to_kleisli_incl_nat_trans_enrichment : nat_trans_enrichment (functor_to_kleisli_cat_incl_nat_trans M) kleisli_incl_enrichment (functor_comp_enrichment (Left_Kleisli_functor_enrichment EM) functor_to_kleisli_cat_enrichment).
proof: Proof. use nat_trans_enrichment_via_comp. intros x y. use (dialgebra_enrichment_mor_eq_of_mor V HV EM (functor_id_enrichment E)). rewrite !assoc'. rewrite kleisli_cat_enrichment_precomp_arr. rewrite kleisli_cat_enrichment_postcomp_arr. rewrite !assoc ; cbn. rewrite precomp_arr_id, postcomp_arr_id. rewrite !id_right. rewrite !assoc'. rewrite functor_to_kleisli_cat_enrichment_data_incl. rewrite free_alg_enrichment_mor_incl. unfold functor_to_kleisli_cat_enrichment_mor. rewrite !assoc. rewrite <- functor_enrichment_postcomp_arr. rewrite !assoc'. rewrite <- postcomp_arr_comp. etrans. { do 2 apply maponpaths. exact (@Monad_law2 _ M y). } rewrite postcomp_arr_id. apply id_right. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Subcategory.Core UniMath.CategoryTheory.Subcategory.Full UniMath.CategoryTheory.Categories.Dialgebras UniMath.CategoryTheory.Categories.EilenbergMoore UniMath.CategoryTheory.Categories.KleisliCategory UniMath.CategoryTheory.EnrichedCats.Enrichment UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation UniMath.CategoryTheory.EnrichedCats.EnrichmentMonad UniMath.CategoryTheory.EnrichedCats.Examples.DialgebraEnriched UniMath.CategoryTheory.EnrichedCats.Examples.EilenbergMooreEnriched UniMath.CategoryTheory.EnrichedCats.Examples.FullSubEnriched UniMath.CategoryTheory.EnrichedCats.Examples.ImageEnriched UniMath.CategoryTheory.EnrichedCats.Examples.KleisliEnriched UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monads.Monads UniMath.CategoryTheory.Monads.KleisliCategory UniMath.CategoryTheory.Limits.Equalizers