Proposition enriched_functor_from_univalent_kleisli_cat_eq (x : C) : pr11 enriched_functor_from_univalent_kleisli_cat_nat_trans (M x) · γ x = # (pr11 enriched_functor_from_univalent_kleisli_cat) (kleisli_nat_trans M x) · pr11 enriched_functor_from_univalent_kleisli_cat_nat_trans x. Proof. cbn -[lift_enriched_functor_along_comm enriched_functor_from_univalent_kleisli_cat]. rewrite !id_right. rewrite !assoc. etrans. { do 2 apply maponpaths_2. refine (_ @ functor_id (pr1 enriched_functor_from_univalent_kleisli_cat) _). apply maponpaths. use eq_mor_kleisli_cat. apply idpath. } etrans. { apply maponpaths_2. apply id_left. } refine (!_). rewrite !assoc'. etrans. { apply maponpaths. etrans. { apply maponpaths_2. refine (_ @ functor_id (pr1 enriched_functor_from_univalent_kleisli_cat) _). apply maponpaths. use eq_mor_kleisli_cat. apply idpath. } apply id_left. } pose (nat_trans_ax (pr11 (lift_enriched_functor_along_comm _ (essentially_surjective_functor_to_kleisli_cat M) (fully_faithful_functor_to_kleisli_cat_enrichment HV EM) LL)) (M x) x (kleisli_monad_nat_trans M x)) as r. refine (_ @ r @ _). - apply maponpaths_2. refine (maponpaths (λ z, # _ z) _). use eq_mor_kleisli_cat. cbn. rewrite functor_id, id_left. apply idpath. - apply maponpaths. cbn. rewrite functor_id, 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.