proposition: Lemma is_nat_trans_etainv (T : Monad C) : is_nat_trans T (kleisli_functor_data (kleislify T)) (eta_arrow T).
proof: Proof. intros a b f. simpl. unfold eta_arrow. rewrite id_right. rewrite id_left. unfold map, bind, η. simpl. unfold Monads.bind, Monads.η, Monads.μ. rewrite functor_comp. rewrite <- assoc. set (H := Monad_law2 (T := T) b). progress simpl in H. unfold Monads.η, Monads.μ in H. etrans. 2: { apply cancel_precomposition. apply pathsinv0. apply H. } now rewrite 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