Re: pushouts in toposes

The distributivity condition is not only necessary and but a sufficient condition for the pushout result. A square A'--> B' | | A --> B f in which the vertical arrows are monic is a pushout iff the following three conditions hold: 1) the square is a pullback; 2) B is the union of Image(f) and B'; 3) the congruence, E, induced by f is the union of the identity relation and E ^ (A' x A'). Hence, the desired result reduces to: B = Image(f) v /\Bi; E = I v /\(E ^ (Ai x Ai)) = I v (E ^ /\(Ai x Ai)).