Given a monad in a 2-category , the Kleisli object of is, if it exists, the universal right -module or -opalgebra. Equivalently, represents the functor that takes an object of to the category of right -modules .
This means that there is a morphism and a 2-cell that induce an isomorphism : given a right -module , there is a unique morphism whose composite with (repsectively ) is equal to (resp. ).
For a monad in the bicategory Prof of profunctors, its Kleisli object consists of a category equipped with a bijective-on-objects functor . The category has the same objects as , with hom-sets . Identities and composition are given by the unit and multiplication of .
Every functor yields a monad in , whose Kleisli object is part of the (bijective on objects, fully-faithful) factorization of .
Because of this, we can identify a monad on in with a bijective-on-objects functor .