class Contravariant f where
  contramap :: (a -> b) -> f b -> f a

-- The flipped internal hom functor
newtype (:^:) a b = Hom { getHom :: b -> a}

-- is contravariant in the second argument
instance Contravariant ((:^:) a) where
  contramap f (Hom g) = Hom $ g . f

-- and covariant in the first, which we'll try to express...

-- This would be nice, but Haskell does not allow unsaturated type families :(
{-
type family Flip (f :: k -> i -> Type) (a :: i) (b :: k) :: Type where
  Flip f a b = f b a

instance Functor (Flip @a @b (:^:) a b) where
  fmap f (Hom g) = Hom (f . g)
-}

-- so we have to resort to rather crude methods...
newtype FHom b a = FHom (a :^: b)

instance Functor (FHom b) where
  fmap f (FHom (Hom g)) = FHom $ Hom (f . g)


instance Semigroup a => Semigroup (a :^: b) where
  (Hom f) <> (Hom g) = Hom $ \x -> f x <> g x

instance Monoid a => Monoid (a :^: b) where
  mempty = Hom (const mempty)