proposition: Lemma etainv_morph_law (T : Monad C) : Monad_Mor_laws (T := T) (T' := (unkleislify (kleislify T))) (etainv_data T).
proof: Proof. split; simpl; intro a. - unfold eta_arrow. rewrite id_right. rewrite id_left. unfold Monads.μ, bind, μ, map, μ, bind, map, μ. simpl. unfold Monads.bind, Monad_Kleisli_data; simpl. unfold Monads.μ, η. rewrite functor_comp. cbn. rewrite functor_id. do 2 rewrite id_left. set (H1 := Monad_law2 (T := T) (T a)). unfold Monads.μ, Monads.η in H1. progress simpl in H1. simpl. unfold Monads.η. etrans. 2: { apply cancel_postcomposition. apply pathsinv0. apply H1. } now rewrite id_left. - apply id_right. Defined.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Monads.Monads UniMath.CategoryTheory.Monads.KTriples UniMath.CategoryTheory.catiso UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.FullyFaithful