Rosebrugh and Wood [Proc AMS 122(2), 409-413, 1994] characterized Set up to equivalence as the only category with a string of four adjoints to the left of its Yoneda embedding. A slew of questions about infinite sets being up for grabs, such as the number of infinite cardinals less than 2^N, it is clear that any such on-the-nose characterization must smuggle in much if not all of what it sets out to characterize. Finite sets do not raise these sorts of questions, removing the above argument for the inevitability of smuggling set-theoretic knowledge into a characterization of FinSet. Now some logicians such as Sol Feferman, and one imagines at least a few category theorists, view category theory as built on set theory. An alternative viewpoint is that the basic notions of category theory exist independently of the category of sets. Where one sits in this spectrum is presumably correlated with how strongly one feels that set theory has been smuggled into the following. For ignorance of the correct name I'll call an object b "strongly indecomposable" when Hom(b,-) preserves binary sums. "Successor object" seems like a reasonable name for an object of the form b+1 (1 the final object). Write FinC for the full subcategory of C whose objects have finitely many elements (morphisms from 1). Claim. Let C be a category with finite sums and final object 1. If 1 is a strongly indecomposable generator and every object is either initial or a successor, then FinC is equivalent to FinSet. (Set, FinSet, and Stone all meet these conditions on C, which could be weakened without changing the conclusion by replacing "finite sums" by "sums with 1" and "strongly indecomposable" (SI) by the requirement that b+1 have exactly one element not an element of b. Or, the dichotomy condition could be strengthened to "For all a,b there exists c such that either a ~ b+c or b ~ a+c.") Proof: Given b in FinC, use dichotomy to shed its n elements sequentially yielding b = 0+1+...+1. SI prevents loss or repetition of any element. For any c (in FinC or not), |0=>c| = 1 since 0 is initial, and by induction on n, |b=>c| >= |c|^n. But 1 generates so |b=>c| <= |c|^n. Pointers to previous appearances of this would be appreciated. Do other familiar categories besides finite sets have a similarly short and elementary characterization, e.g. finite Boolean algebras, finite Abelian groups, finite posets, finite monoids, etc? Finite Boolean algebras are easily characterized as dual finite sets. Specifically, for any category C with finite products and initial object 2, if 2 is a strongly coindecomposable cogenerator and every object is either final or of the form 2xb, then FinC is equivalent to FinBool. Bool, FinBool, and CABA all satisfy these conditions on C. ("2 strongly coindecomposable" means of course that Hom(-,2) sends finite products to finite sums; equivalently, every predicate q:axb->2 on axb factors through exactly one of the projections p1:axb->a, p2:axb->b, i.e. it acts as a predicate on one of a or b and is constant on the other.) Have we smuggled Boolean algebra into this argument? Is category theory based on set theory to any greater extent than on Boolean algebra? And is there any essential difference between this argument and its mate for FinSet? Sets and Boolean algebras would seem to deserve equal credit for their foundational role in both. My own view, first articulated in LICS'95, 444-454, is that mathematics is a catenary, the "Stone Gamut," held up at each end by the twin categories Set and CABA. The catenary is created from their interaction. Any suggestions for characterizing finite Abelian groups? Vaughan Pratt