Proposition enriched_nat_trans_from_univalent_kleisli_cat_help_eq (x : C) : # (functor_to_kleisli_cat M ∙ pr1 G₁) (kleisli_monad_nat_trans M x) · pr11 enriched_nat_trans_from_univalent_kleisli_cat_help x = pr11 enriched_nat_trans_from_univalent_kleisli_cat_help (M x) · # (functor_to_kleisli_cat M ∙ pr1 G₂) (kleisli_monad_nat_trans M x). Proof. refine (_ @ p x @ _). - cbn. rewrite !assoc. rewrite !id_right. rewrite <- (functor_comp (pr1 G₁)). etrans. { apply maponpaths. refine (_ @ functor_id (pr1 G₂) _). apply maponpaths. use subtypePath ; [ intro ; apply isapropunit | ]. use eq_mor_eilenberg_moore. apply idpath. } rewrite id_right. apply maponpaths_2. apply maponpaths. use eq_mor_kleisli_cat. cbn. rewrite functor_id. rewrite id_left, id_right. apply idpath. - cbn. rewrite !assoc'. rewrite !id_left. refine (!_). etrans. { apply maponpaths_2. refine (_ @ functor_id (pr1 G₁) _). apply maponpaths. use eq_mor_kleisli_cat. apply idpath. } rewrite id_left. rewrite <- (functor_comp (pr1 G₂)). do 2 apply maponpaths. use eq_mor_kleisli_cat. cbn. rewrite functor_id. rewrite !id_left. apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.whiskering. 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.KleisliEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.UnivalentKleisliEnriched. Require Import UniMath.CategoryTheory.EnrichedCats.Examples.FunctorCategory. Require Import UniMath.CategoryTheory.EnrichedCats.RezkCompletion.RezkUniversalProperty. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Monads.KleisliCategory. Require Import UniMath.CategoryTheory.Limits.Equalizers. Section FunctorFromUnivalentKleisli. End FunctorFromUnivalentKleisli. Section NatTransFromUnivalentKleisli. End NatTransFromUnivalentKleisli.