Let $T\_1$ and $T\_2$ be two Grothendieck topologies on the same small category $C$, and let $T\_3 = T\_1 \cup T\_2$ (by which I mean the smallest Grothendieck topology on $C$ containing $T\_1$ and $T\_2$). If $X$ is a $T\_1$-sheaf, is its $T\_2$-sheafification still a $T\_1$-sheaf (and therefore a $T\_3$-sheaf)?


If the answer is "not always," then are there conditions one can impose on $T\_1$ and $T\_2$ to make it true? Does it matter if $X$ is already $T\_2$-separated?


(I'm really interested in the analogous question for stacks, but I'm guessing the answers will be pretty much the same.)



Does sheafification preserve sheaves for a different topology?
I think my answer to [this question](https://mathoverflow.net/questions/6592/localizing-model-structures) provides a counterexample: Let C be the category a → b, and consider the topologies T1 generated by the single covering family {a → b} and T2 generated by declaring the empty family to be a covering of a. (Caution: In my other answer I used covariant functors for some reason, so I hope I didn't err in translating the example.)


Note that if I switch T1 and T2, though, the condition you ask for is satisfied, even though neither class of sheaves contains the other. So there may be some interesting conditions under which it is true.


