The construction of the Kleisli category can be turned into a left adjoint functor whose domain is the category of Mnd monads (on arbitrary categories) and monad morphisms. However, the codomain of this functor is not Cat, but a category AbsKl whose objects I call "abstract Kleisli categories". An abstract Kleisli category is a category K together with a functor L:K->K, a natural transformation \epsilon: L->Id, and a (not generally natural) transformation \theta:Id->L, such that (1) \theta_L is a natural transformation (2) L\theta o \theta = \theta_L o \theta (3) \epsilon o \theta = id (3) L\epsilon o \theta_L = id A morphism K->K' of AbsKl is a functor that preserves the solutions of the non-naturality square \theta o f = Lf o \theta (I) The Kleisli construction forms a functor Mnd->AbsKl. Its right adjoint sends an abstract Kleisli category K to the evident monad on the subcategory given by the solutions of Equation (I). The counit of this adjunction is in fact an iso, so AbsKl is a full reflective subcategory of Mnd. The full subcategory of Mnd which is equivalent to Abskl is given by those monads for which every component of the unit is a regular mono. Cheers, Carsten