Order-adjoint monads and injective objects

Given a monad TT on Set whose functor factors through the category of ordered sets with left adjoint maps, the category of Kleisli monoids   is defined as the category of monoids in the hom-sets of the Kleisli category of TT. The Eilenberg–Moore category of TT is shown to be strictly monadic over the category of Kleisli monoids. If the Kleisli category of TT moreover forms an order-enriched category, then the monad induced by the new situation is Kock–Zöberlein. Injective objects in the category of Kleisli monoids with respect to the class of initial morphisms then characterize the objects of the Eilenberg–Moore category of TT, a fact that allows us to recuperate a number of known results, and present some new ones.