Can a monad be a comonad?

Question

I know what a monad is. I think I have correctly wrapped my mind around what a comonad is. (Or rather, what one is seems simple enough; the tricky part is

Answer 1

Yes. Turning some comments into an answer:

newtype Identity a = Identity {runIdenity :: a} deriving Functor
instance Monad Identity where
  return = Identity
  join = runIdentity
instance CoMonad Identity where
  coreturn = runIdentity
  cojoin = Identity

Reader and Writer are exact duals, as shown by

class CoMonoid m where
  comempty :: (m,a) -> a
  comappend :: m -> (m,m)
--every haskell type is a CoMonoid
--that is because CCCs are boring!

instance Monoid a => Monad ((,) a) where
  return x = (mempty,x)
  join (a,(b,x)) = (a <> b, x)
instance CoMonoid a => CoMonad ((,) a) where
  coreturn = comempty
  cojoin = associate . first comappend

instance CoMonoid a => Monad ((->) a) where
  return = flip (curry comempty)
  join f = uncurry f . comappend
instance Monoid a => CoMonad ((->) a)  where
  coreturn f = f mempty
  cojoin f a b = f (a <> b)