proposition: Lemma etainv_data_laws (T : Monad C) : @Monad_Mor_laws C T (@unkleislify C (@kleislify C T)) (etainv_data T).
proof: Proof. split; simpl; intros. - unfold eta_arrow. rewrite id_right. unfold map, μ, Monads.μ, bind. simpl. unfold Monads.bind, η, Monads.μ. simpl. unfold Monads.η. do 2 rewrite id_left. rewrite functor_id, id_left. rewrite <- assoc. set (H := Monad_law3 (T := T) a). progress simpl in H. unfold Monads.μ in H. etrans. 2: { apply cancel_precomposition. apply H. } rewrite assoc. rewrite <- functor_comp. set (H1 := Monad_law1 (T := T) a). progress simpl in H1. unfold Monads.η, Monads.μ in H1. etrans. 2: { apply cancel_postcomposition. apply maponpaths. apply pathsinv0. eapply H1. } now rewrite functor_id, 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