Since Vaughan insists on doing these things in terms of matrices and we benighted mathematicians insist on thinking in terms of pairings, I thought it might be helpful to describe what he did in those terms. First off, he erred in writing Chu instead of chu. So it is the separated extensional part. I will describe it for chu(Set,2), since nothing much changes for other values of K except you get more dinats. I will describe three types of objects. If A = (A_1,A_2) is an object, I will say that A_1 is the set of points and A_2 the set of states and treat A_2 as a set of subsets of A_1. I will say that A is type I if there is a point in no state AND if the empty set is a state. I will say that A is of type II if there is a point in every state and if the whole of A_1 is a state. These would appear to be quite different, but they are exchanged by the non-trivial automorphism of 2, which means they have the same properties. All remaining objects will be of type III. The first observation is that the type is invariant under formation of A -o A. That is, A -o A has the same type as A. Second, if A and B are of different types, then at least one of Hom(A,B) and Hom(B,A) is empty. This implies that if there is a map A --> B, then B -o A can only be (0,1) or (0,0) (0 is the empty set). That is is either initial or a quotient of the initial object. In either case, it has at most one arrow to any other object and any diagram starting with it commutes. The result of this is that any naturality condition that involves objects of different types is automatic. Either there is no map A --> B to test or there is one and the diagram is automatically commutative. For an object of type I, there is a zero map that takes every point to the (unique!) point not in any state and takes every state to the empty state. This is natural restricted to the type I objects and there is a similar 0 map for type II's. Now there are at least four dinatural endomorphisms of the functor that takes A and B to A -o B. They are all the identity on type III objects and can be either the identity or the 0 map on type I and the same on type II. Michael