I was inspired by George Pollard's post at haskell-cafe and tried to implement the non-polymorphic Functor class ( I named it Functor' ). I changed some names and added reasonable constraints.

    type family NewPt f a
    class Functor' f where
    type Point f
map ∷ (a ~ Point f, b ~ Point g, g ~ NewPt f b, Functor' g) ⇒ (a → b) → f → g

I would like to be able to write:

    type OldPt f = NewPt f (Point f)
    class (f ~ OldPt f) ⇒ Functor' f ...

but ghc says it's not implemented yet (version 6.12.1). However, it's not the main problem.

Now I can write some instances:

    type instance NewPt [a] b = [b]
    instance Functor' [a] where
      type Point [a] = a
      map = fmap

    type instance NewPt ByteString a = ByteString
    instance Functor' ByteString where
      type Point ByteString = Word8
      map = BS.map

But I can't write instance for Set:

    type instance NewPt (Set a) b = Set b
    instance Ord a ⇒ Functor' (Set a) where
type Point (Set a) = a
map = Set.map

ghci complains: Could not deduce (Ord a1) from the context (g ~ NewPt (Set a) a1, a1 ~ Point g, Functor' g)
arising from a use of `Set.map' at ...

The type of Set.map is

    Set.map :: (Ord a, Ord b) => (a -> b) -> Set a -> Set b

(Ord a) is in the instance context, and what about b? Type of map for Set instance would be:

original:
map ∷ (a ~ Point f, b ~ Point g, g ~ NewPt f b, Functor' g) ⇒ (a → b) → f → g

substitute: f → Set a, g → Set b
    map :: Functor' (Set b) ⇒ (a →b) →Set a →Set b

(Ord b) must be deduced from (Functor (Set b)) but it doesn't. I don't know whether it's my mistake somewhere or ghc problem.

(Sorry for my English, it's not perfect).
--
All the best,
Alexey