Let E be a topos then Fam(E) -> Set is certainly a fibered topos but by Th.6.2.3 of Pieter Hofstra's Thesis Fam(E) is a topos iff E is an atomic category (in the sense of Johnstone's 1977 book on Topos Theory, exercise 12 on p. 257). But in atomic categories all morphisms are epic and thus Fam(E) is a topos only if E is trivial.
Alas, there is a flaw in Pieter's Th.6.2.3 (which certainly is not crucial for the main results of his otherwise very nice Thesis). Actually, it can be seen quite easily: if E is a cocomplete topos then Fam(E) is equivalent to the glueing of Delta : Set -> E which is known to be a topos. So it seems to be open to characterize those toposes E for which Fam(E) is a topos. In particular, I don't know the answer for E the free topos (with nno) or a realizability topos. In the latter case we know that glueing of Nabla (right adjoint to Gamma) is a topos but it's different from Fam(E). I'd be grateful about any suggestions even for these particular cases! Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]