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.)