  If the ultrafilter U is nonprincipal, i.e. contains no finite sets, then the diagonal map is not onto the domain of U-prod A, and in fact U-prod A is generally much larger than A. So we have a way of constructing large elementary extensions. The axiom of choice guarantees that every infinite set has many nonprincipal ultrafilters over it. Ultrapowers are an essential tool for handling large cardinals in set theory (see the entry on  set theory).