When is the diagonal functor Frobenius?

Given a complete, cocomplete category C{script}, we investigate the problem of describing those small categories I such that the diagonal functor {increment}: C{script} → Functors(I,C{script}) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → C{script} are naturally isomorphic. We find necessary conditions on I for a certain class of categories C{script}, and, as an application, we give both necessary and sufficient conditions in the two special cases C{script} = Set or _RM, the category of left modules over a ring R.