I used the phrase regular--or effective--epimorphic family for the dual notion of pair of arrows that were a pushout. Of course, this is a much wider notion, so regular epi pair would do for that. Since it is almost self-defining and also shorter than some of the alternatives, it is the term I recommend. Richard points out, in effect, that a subset R in A x B is a pullback in this sense iff R o R\op o R is in R. That is true in any topos. It is also true in any abelian category. Is it true in the dual of a topos? In the dual of sets? Whatever, it is an exactness condition (that any such subset be a pullback) that might turn out to be interesting. Conversely, it could turn out to be equivalent (probably with some existence of limits and/or colimits) to effective equivalence relations. Note that it implies that R o R\op and R\op o R are transitive (they are obviously symmetric). Michael ++++++++++++++++++++++++