proposition: Proposition kleisli_nat_trans_enrichment : nat_trans_enrichment (kleisli_nat_trans M) (functor_comp_enrichment EM kleisli_incl_enrichment) kleisli_incl_enrichment.
proof: Proof. use nat_trans_enrichment_via_comp. intros x y ; cbn. use (dialgebra_enrichment_mor_eq_of_mor V HV EM (functor_id_enrichment _)). rewrite !assoc'. etrans. { apply maponpaths. exact (kleisli_cat_enrichment_precomp_arr (kleisli_incl M y) (kleisli_nat_trans M x)). } etrans. { rewrite !assoc. apply maponpaths_2. apply free_alg_enrichment_mor_incl. } refine (!_). etrans. { do 2 apply maponpaths. exact (kleisli_cat_enrichment_postcomp_arr (kleisli_incl M (M x)) (kleisli_nat_trans M y)). } etrans. { apply maponpaths. rewrite !assoc. apply maponpaths_2. apply free_alg_enrichment_mor_incl. } cbn. rewrite !assoc. exact (!(nat_trans_enrichment_to_comp (mu_of_monad_enrichment EM) x y)). 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