There's a well known theorem due to Beck that characterizes when an adjunction is monadic, that is, if $F$ is left adjoint to $G$, $G:D \to C$, $GF:=T$ is always a monad on $C$, and the adjunction is called monadic, essentially, when $D$ is the Eilenberg–Moore category $C^T$ of $T$-algebras and $G$ is the forgetful functor. (For the precise definition see http://ncatlab.org/nlab/show/monadic+adjunction). I was wondering if there was a similar characterization to determine when $D$ is the Kleisli category of FREE $T$-algebras?