Therefore, an arrow can be thought as a putative replacement of . In fact an arrow comes equipped with a transformation that lift morphisms of to the ‘augmented’ morphisms given by , and with a transformation that behaves like a composition operation. Then one requires that is associative and that is the unit of this operation.
These data makes a promonad? on , i.e. a monad on in the bicategory of profunctors. Since every monad on (or on any reasonable enriching base where programmers work) is strong, arrows traditionally generalize the hom-profunctor of strong monads. The additional requirement that is a strong profunctor (and the laws it is required to satisfy) imply that an arrow is a strong monad in the bicategory of profunctors (Asada10).
Sam Lindley?, Philip Wadler, Jeremy Yallop?, Idioms are Oblivious, Arrows are Meticulous, Monads are Promiscuous, Electron. Notes Theor. Comput. Sci. 229(5), pp. 97–117, 2011 (doi:10.1016/j.entcs.2011.02.018)