is union associative?

14 Jan 1995

      I suspect that the following question is very natural and
a known solution has to exist, but I cannot find any reference
in books I have.

Consider the following diagram

	A
	|a
	V    b
	W <----- B

and define union of two morphisms a U b := Pushout(Pullback(X))

In the category Set this operation is obviously associative:
for any arrows a,b,c with common target,

	a U (b U c) = (a U b) U c ,

Can anybody tell,  what conditions should be imposed to a
category to guarantee this associativity?

Thank you in advance,
Ilya Beylin