proposition: Lemma is_inverse_etainv (T : Monad C) : is_inverse_in_precat (eta_morph T : category_Monad C ⟦unkleislify (kleislify T), T⟧) (etainv_morph T : category_Monad C ⟦T, unkleislify (kleislify T)⟧).
proof: Proof. split. - apply Monad_Mor_eq. intros. simpl. unfold eta_arrow. apply id_left. - apply Monad_Mor_eq. intros. simpl. unfold eta_arrow. apply id_left. 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