Proposition pre_whisker_enriched_nat_trans_from_univalent_kleisli_cat (x : C) : pr11 (pre_whisker_enriched_functor_category K enriched_nat_trans_from_univalent_kleisli_cat) x = pr11 α x. Proof. pose (maponpaths (λ z, pr11 z x) (lift_enriched_nat_trans_along_comm _ (essentially_surjective_functor_to_kleisli_cat M) (fully_faithful_functor_to_kleisli_cat_enrichment HV EM) ζζ)) as q. refine (q @ _). cbn. rewrite !id_left, !id_right. etrans. { apply maponpaths. refine (_ @ functor_id (pr1 G₂) _). apply maponpaths. use eq_mor_kleisli_cat. apply idpath. } refine (id_right _ @ _). etrans. { apply maponpaths_2. refine (_ @ functor_id (pr1 G₁) _). apply maponpaths. use eq_mor_kleisli_cat. apply idpath. } apply id_left. 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.