Suppose we have a Grothendieck pretopology $\tau$ on a category C with fibered products. Now define a new Grothendieck pretopology $\tau'$ consisting of all families of morphisms refinable by $\tau$-covers. That is, the new covers are the families $\{V_\beta \to X\}$ such that there exists some $\tau$-cover $\{U_\alpha \to X\}$ and a factorisation $U_\alpha \to V_{\beta_\alpha} \to X$ for each $\alpha$. This new set of families is also a Grothendieck pretopology and the question is: do they give the same topos? That is, is a presheaf a $\tau$-sheaf if and only if it is a $\tau'$-sheaf?

Edit: I could't read the relevant page in Elephant either, but Mike's answer lead me to the saturation section of http://ncatlab.org/nlab/show/coverage after which I worked out how to prove it myself. If someone explains to me how to typeset diagrams, I'll write up the answer.