Lemma is_inverse_epsinv (T : KleisliMonad C) : is_inverse_in_precat (eps_morph T : category_Kleisli C ⟦T, kleislify (unkleislify T)⟧) (epsinv_morph T). Proof. split. - apply Kleisli_Mor_eq. apply funextsec. intro a. simpl. unfold eps. apply id_left. - apply Kleisli_Mor_eq. apply funextsec. intro a. simpl. unfold eps. apply id_left. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Monads.Monads. Require Import UniMath.CategoryTheory.Monads.KTriples. Require Import UniMath.CategoryTheory.catiso. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.FullyFaithful.