Proposition kleisli_nat_trans_enrichment : nat_trans_enrichment (kleisli_nat_trans M) (functor_comp_enrichment EM kleisli_incl_enrichment) kleisli_incl_enrichment. 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.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Subcategory.Core. Require Import UniMath.CategoryTheory.Subcategory.Full. Require Import UniMath.CategoryTheory.Categories.Dialgebras. Require Import UniMath.CategoryTheory.Categories.EilenbergMoore. Require Import UniMath.CategoryTheory.Categories.KleisliCategory. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentMonad. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.DialgebraEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.EilenbergMooreEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.FullSubEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.ImageEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.KleisliEnriched. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Monads.KleisliCategory. Require Import UniMath.CategoryTheory.Limits.Equalizers.