Functor module:Data package:profunctors

For a good explanation of profunctors in Haskell see Dan Piponi's article: http://blog.sigfpe.com/2011/07/profunctors-in-haskell.html For more information on strength and costrength, see: http://comonad.com/reader/2008/deriving-strength-from-laziness/

Formally, the class Profunctor represents a profunctor from Hask -> Hask. Intuitively it is a bifunctor where the first argument is contravariant and the second argument is covariant. You can define a Profunctor by either defining dimap or by defining both lmap and rmap. If you supply dimap, you should ensure that:

dimap id id ≡ id

If you supply lmap and rmap, ensure:

lmap id ≡ id
rmap id ≡ id

If you supply both, you should also ensure:

dimap f g ≡ lmap f . rmap g

These ensure by parametricity:

dimap (f . g) (h . i) ≡ dimap g h . dimap f i
lmap (f . g) ≡ lmap g . lmap f
rmap (f . g) ≡ rmap f . rmap g

Laws:

unit . counit ≡ id
counit . unit ≡ id

Laws:

proextract . promap f ≡ f . proextract
proextract . produplicate ≡ id
promap proextract . produplicate ≡ id
produplicate . produplicate ≡ promap produplicate . produplicate

ProfunctorFunctor has a polymorphic kind since 5.6.

Laws:

promap f . proreturn ≡ proreturn . f
projoin . proreturn ≡ id
projoin . promap proreturn ≡ id
projoin . projoin ≡ projoin . promap projoin