When is the Diagonal Functor Frobenius?

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