Dear Categorists - Mike Stay's puzzle is quite nice. I won't answer it. Instead, I will generalize and extend it. Suppose we have a category C with finitely many objects and morphisms. Let C^ be the category of presheaves on C, and Omega the subobject classifier in C^. There is a Lawvere theory T whose n-ary operations are the morphisms from Omega^n to Omega. 1. Is T finitely generated? That is, can we find a finite collection of morphisms f: Omega^n -> Omega (for various choices of n) such that T is generated, as a category with finite products, by these? 2. Can we find an explicit set of generators for T? 3. Is T finitely presented? 4. Can we find an explicit set of generators and relations for T? My guess on 3 is "yes". Mike was interested in the case where C^ is the category of reflexive graphs, but I think the question would be interesting to study in detail for lots of other classic examples. Someone must have done it already, no? Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]