Peter Johnstone is right -- the monad David described doesn't exist, thus neither does the monad morphism I described in my other post (... sorry!). Perhaps the fibrations monad is still of interest. One could fix a skeleton of Set_k, and for k = cardinality of natural numbers this works fine, and the monad on Set you get is the monoid monad. However for bigger k you're likely to run into problems when trying to do this sort of thing. However it isn't true that the monad
(... on Cat, where we replace k-Set with k-Cat.)
is the k-coproduct completion monad -- you need to keep k-Set but work in Cat and take lax slices, ie take the monad on Cat which has underlying endofunctor X |-> k-Set // X (where // means "lax slice") to get the k-coproduct completion monad. Mark Weber [For admin and other information see: http://www.mta.ca/~cat-dist/ ]