proposition: 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: 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.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Categories.EilenbergMoore UniMath.CategoryTheory.Categories.KleisliCategory UniMath.CategoryTheory.EnrichedCats.Enrichment UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation UniMath.CategoryTheory.EnrichedCats.EnrichmentMonad UniMath.CategoryTheory.EnrichedCats.Examples.KleisliEnriched UniMath.CategoryTheory.EnrichedCats.Examples.UnivalentKleisliEnriched UniMath.CategoryTheory.EnrichedCats.Examples.FunctorCategory UniMath.CategoryTheory.EnrichedCats.RezkCompletion.RezkUniversalProperty UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monads.Monads UniMath.CategoryTheory.Monads.KleisliCategory UniMath.CategoryTheory.Limits.Equalizers