So the equations for these cups and caps look awfully like one of the ways we saw of defining Galois connections (and then adjoint functors), namely with id->R.L and L.R->id. To make this precise, is the category of endofunctors (on certain categories) monoidal (and compact - if that's the right word for having the cup and cap), with the above natural transformations as the cup and cap, id as the unit, and *something* as the tensor? Oh wait, the tensor is monadic *join*, am I talking about monads by accident?
Edit: no, that's not quite right - I'm confusing a monoid in a category with a monoidal category. Hmmm