Generalization of strong and closed profunctors

Question

I was looking at the classes of strong and closed profunctors:

class Profunctor p where
    dimap :: (a' -> a) -> (b -> b') -> p a b -> p a' b'
class Profunctor p => Strong p where
    strong :: p a b -> p (c, a) (c, b)
class Profunctor p => Closed p where
    closed :: p a b -> p (c -> a) (c -> b)

((,) is a symmetric bifunctor, so it's equivalent to the definition in "profunctors" package.)

I note both (->) a and (,) a are endofunctors. It seems Strong and Closed have a similar form:

class (Functor f, Profunctor p) => C f p where
    c :: p a b -> p (f a) (f b)

Indeed, if we look at the laws, some also have a similar form:

strong . strong ≡ dimap unassoc assoc . strong
closed . closed ≡ dimap uncurry curry . closed

lmap (first f) . strong ≡ rmap (first f) . strong
lmap (. f)     . closed ≡ rmap (. f)     . closed

Are these both special cases of some general case?

Answer 1

You could add Choice to the list. Both Strong and Choice (or cartesian and cocartesian, as Jeremy Gibbons calls them) are examples of Tambara modules. I talk about the general pattern that includes Closed in my blog post on profunctor optics (skip to the Discussion section), under the name Related.