applicative.hs

-- same as sum below
liftSum :: Functor f => Either (f a) (f b) -> f (Either a b)
liftSum = either (Left <$>) (Right <$>)

-- same as comult below
unliftProd :: Functor f => f (a, b) -> (f a, f b)
unliftProd x = (fst <$> x, snd <$> x)

-- same as cosum below
unliftSum :: Comonad f => f (Either a b) -> Either (f a) (f b)
unliftSum x = either (\y -> Left $ fmap (either id (const y)) x)
                     (\y -> Right $ fmap (either (const y) id) x)
                     (extract x)

-- could be Monad to make it "more" dual to unliftSum
-- liftProd :: Monad f => (f a, f b) -> f (a, b)
-- same as mult below
liftProd :: Applicative f => (f a, f b) -> f (a, b)
liftProd (x, y) = (,) <$> x <*> y

applicative2.hs

-- same as applicative
-- strong lax monoidal functor
class Functor f => Monoidal f where
  one :: () -> f ()
  mult :: (f a, f b) -> f (a, b)

-- oplax monoidal or lax comonoidal functor
class Functor f => CoMonoidal f where
  coone :: f () -> ()
  comult :: f (a, b) -> (f a, f b)

-- strong lax monoidal functor
class Functor f => Monoidal' f where
  zero :: Void -> f Void
  sum :: Either (f a) (f b) -> f (Either a b)

-- oplax monoidal or lax comonoidal functor
class Functor f => CoMonoidal' f where
  cozero :: f Void -> Void
  cosum :: f (Either a b) -> Either (f a) (f b)