Is there a term for a monad that is also a comonad?

Question

I'm just wondering whether there's a concise term for something that's both a monad and a comonad. I've done some searching, and I know these structures exist, but I haven't found a name for them.

Answer 1

As far as I know, there is no term to define it because a monad-comonad would enforce nothing: you can always do a return to get in or an extract to get out.

As types are there to enforce some constraints, a too permissive constraint wouldn't be of any use. As no one would use it (except for the identity), no one probably bothered to name it.

Answer 2

Such a creature, subject to certain conditions, is sometimes called a "Hopf monad" or a "Bimonad" (http://ncatlab.org/nlab/show/Hopf+monad).

However, this also requires fulfilling a number of axioms regarding distributive properties, and I haven't seen it come up in a programming context in any particular way.