Let me suggest still another terminology: For this, call a class S of maps in an arbitrary category *(co)stable* iff S is closed under composition and under (co)base change. Then call a commutative square *S-exact * (resp. *S-coexact*) iff the induced map to the pullback (resp. from the pushout) belongs to S. It is then easy to check that S-(co)exact squares compose for any (co)stable class S (which I believe is the minimal condition to impose on any reasonable distinguished class of commutative squares). In an abelian category, the class M of monos (resp. the class E of epis) is not only stable (resp. costable) but also costable (resp. stable). With this terminology, Hilton's exact squares can either be identified with the E-exact squares or with the M-coexact squares, which explains why it is a self-dual concept, cf. the first message of Michael Barr and the last message of Marco Grandis. In homotopy theory, there is the important concept of a *homotopy pullback* which is the ``homotopy invariant'' substitute for an ordinary pullback. For those who are familiar with Quillen model categories, it is very useful in practice that if a Quillen model category is *right proper* (i.e. its class of fibrations is stable), then a commutative square with two parallel fibrations is a homotopy pullback *if and only if* the square is exact with respect to the class of trivial fibrations (those fibrations which are also weak equivalences). There is of course a dual statement for homotopy pushouts in a left proper Quillen model category. With best regards, Clemens Berger.