1. In an abelian category, I would prefer "exact", or "Hilton-exact". Hilton considered such squares (for abelian categories), and proved =20 that an equivalent condition is that this square (of proper =20 morphisms) is "bicommutative" in the category of relations (i.e. it =20 commutes and stays commutative when you reverse two "parallel" arrows =20= - as relations). Plainly: bicartesian square =3D> pullback =3D> exact; and dually. REFERENCE: P. Hilton, Correspondences and exact squares, in: Proc. Conf. on =20 Categorical Algebra, La Jolla 1965, Springer, pp. 254-271. 2. Studying more general categories of relations, I considered =20 "semicartesian squares" (f,g, h,k), defined - in any category - by =20 the following self-dual property (after being commutative, of course): Whenever (f',g', h,k) and (f,g, h',k') commute, also the =20 external square (f',g', h',k') commutes B f' f h h' A' A D D' g' g k k' C (add slanting arrows f': A' --> B, f: A --> B, etc). - Again: bicartesian square =3D> pullback =3D> semicartesian, and =20= dually. - If pb's and/or po's exist, one can give a lot of equivalent =20 properties; eg: -- (f,g) and the pb of (h,k) have the same po (or the same =20 commutative squares out of them). - In an abelian category, semicartesian amounts to the previous notion. - In Set, it characterises again those squares which are =20 bicommutative in Rel. REFERENCE: M. Grandis, Sym=E9trisations de categories et factorisations =20 quaternaires, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. =20 14 sez. 1 (1977), 133-207. 3. A 2-dimensional version of this property (actually a STRUCTURE on =20 2-cells), was introduced by Guitart, and called "H-exact", if I =20 remember well (H for Hilton) REFERENCES: - R. Guitart, Carr=E9s exacts et carr=E9s deductifs, Diagrammes 6 = (1981), =20 G1-G17. - R. Guitart and L. Van den Bril, Calcul des satellites et =20 pr=E9sentations des bimodules =E0 l'aide des carr=E9s exacts, Cahiers =20= Topologie G=E9om. Diff=E9rentielle 24 (1983), no. 3, 299-330. (and some other papers by the same authors). Best regards Marco Grandis