proposition: 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: 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.
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