Proposition Kleisli_cat_enrichment_postcomp_arr {x y : Kleisli_cat_monad M} (w : Kleisli_cat_monad M) (f : x --> y) : postcomp_arr Kleisli_cat_monad_enrichment w f = postcomp_arr E w (#M f · μ M y). Proof. unfold postcomp_arr ; cbn. rewrite !assoc'. apply maponpaths. rewrite enriched_comp_postcomp_arr. rewrite !assoc. apply maponpaths_2. rewrite <- !tensor_comp_id_r. apply maponpaths_2. rewrite <- functor_enrichment_from_arr. rewrite enriched_from_arr_postcomp. apply idpath. 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.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentMonad. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Monads.KleisliCategory. Section FunctorFromKleisli. End FunctorFromKleisli.