The Kleisli category of a monad on has the same objects as , but a morphism is a -map , a “-effectful” arrow into the monad. The identity is the unit , and the composite of with is (written in diagram order). It is the category of free-algebras and sits in the canonical adjunction whose induced monad is . The prototype is , the markov-category whose morphisms are Markov kernels. Likewise and is the partial functions. Kleisli composition is exactly the semantics of sequencing effectful programs.
References
Heinrich Kleisli, “Every standard construction is induced by a pair of adjoint functors”, Proc. Amer. Math. Soc. 16 (1965)
Michele Giry, “A categorical approach to probability theory” (1982)