Definition. Kleisli category [dm-000T]

For a monad (\mathcal {M}, \mu , \eta ) over the category \mathbf {C}, the Kleisli category has the same objects as \mathbf {C}, but the morphisms A \xrightarrow {h} B are those morphisms h of \mathbf {C} that have the form A \xrightarrow {h} \mathcal {M}(B). Composition is defined as follows, for morphisms A \xrightarrow {f} \mathcal {M}(B) and B \xrightarrow {g} \mathcal {M}(C): g \circ _K f = \mu _C \circ \mathcal {M}(g) \circ f Identity morphisms for this category are just \eta .