Apparently, every Arrow is a Strong profunctor. Indeed ^>> and >>^ correspond to lmap and rmap . And first' and second' are just the same as first and second . Similarly every ArrowChoice is also Choice . What profunctors lack compared to arrows is the ability to compose them. If we add composition, will we get an arrow? In other words, if a (strong) profunctor is also a category , is it already an arrow? If not, what's missing? What profunctors lack compared to arrows is the ability to compose them. If we add composition, will we get an arrow? MONOIDS This is exactly the question tackled in