As Peter Johnstone also emphasized in his reply, the construction which David Spivak describes, namely "T(S)=(k-Set \downarrow {S})", is really a part of a well known "monad" on the category of categories: if S is any category, T(S) is the free cocompletion of S under k-small coproducts. It is only a monad up to canonical isomorphisms, because coproducts are not in general strictly associative. This cocompletion "monad" under coproducts has been widely studied under the name "Fam" (because T(S) is the category of k-small Families of objects in S). It is an example of a KZ monad. However, replacing k-Set by k-Cat provides a monad on Cat which is not KZ; David observes: "(There is a similar monad on Cat, where we replace k-Set with k-Cat.)" and Peter's reply to this: "This is correct, and it's well-known: it is the monad which freely adjoins k-small coproducts to a category. " does not apply here (it slipped into the wrong place of his reply): rather, David's "similar monad" is trying to provide free cocompletion under colimits indexed by k-small categories, but does not, until you make a category-of-fractions construction on its values. My University of Chicago thesis (1967) described this way of making free cocompletions. This "similar monad" (before doing the fractions-part) has been studied by Guitart, he calls it this monad DIAG. Reference: Guitart, René, Remarques sur les machines et les structures. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 15 no. 2 (1974), p. 113-144 (available electronically in NUMDAM). Anders Kock [For admin and other information see: http://www.mta.ca/~cat-dist/ ]