What's the relationship between profunctors and arrows?

Question

Apparently, every Arrow is a Strong profunctor. Indeed ^>> and >>^ correspond to lmap and rmap. And fir

Answer 1

@haoformayor's answer (and the referenced paper) is a great insight into the underlying category theory - monoidal categories are rather beautiful! - but I thought some code showing you how to turn an Arrow into a Strong Category and vice versa as they appear in their respective libraries might make a helpful addendum.

import Control.Arrow
import Control.Category
import Data.Profunctor
import Data.Profunctor.Strong
import Prelude hiding (id, (.))

One way...

newtype WrapP p a b = WrapP { unwrapP :: p a b }

instance Category p => Category (WrapP p) where
    id = WrapP id
    WrapP p . WrapP q = WrapP (p . q)

instance (Category p, Strong p) => Arrow (WrapP p) where
    first = WrapP . first' . unwrapP
    second = WrapP . second' . unwrapP

    -- NB. the first usage of id comes from (->)'s Category instance (id :: a -> a)
    -- but the second uses p's instance (id :: p a a)
    arr f = WrapP $ dimap f id id

... and t'other...

newtype WrapA p a b = WrapA { unwrapA :: p a b }

instance Arrow p => Profunctor (WrapA p) where
    dimap f g p = WrapA $ arr f >>> unwrapA p >>> arr g

instance Arrow p => Strong (WrapA p) where
    first' = WrapA . first . unwrapA
    second' = WrapA . second . unwrapA