Lemma is_catiso : is_catiso (functor_unkleislify(C:=C)). Proof. split. - apply fully_faithful_from_equivalence. use (is_left_adjoint_functor_unkleislify,, form_equivalence_unkleislify). - apply (isweq_iso _ (λ T : Monad C, kleislify T)). + intro T. simpl. apply kleislify_unkleislify. + simpl. apply unkleislify_kleislify. 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.