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