Does sheafification preserve sheaves for a different topology? 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.)
 A: I think my answer to this question 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.
A: I am not sure whether I will answer your question properly. I just tell some facts which might be useful
I know there are following categories equivalence
Qcoh(X),Qcoh(X,t),Qcoh(aX,t)
X is a presheaf
Qcoh(X) is quasi coherent modules on X,t is a grothendieck topology. aX is the sheafification according to topology t of X.
If t is coarser than topology of effective descent, then those three categories are equivalent. Which means Qcoh(X)=Qcoh(X,t1)=Qcoh(X,t2)=Qcoh(aX,t1)=Qcoh(aX,t2) are the same if t1 t2 are coarser than effective descent topology. This fact shows that it is not necessary to consider sheaf but presheaf. Their descent theory can be described by category of quasi coherent sheaves.
Similar results applied to stack. These facts are indicated in Giraud's book,but not wrote it out. They were proved by Orlov in his paper quasi coherent sheaves in commutative and noncommutative geometry and Kontsevich-Rosenberg preprint in MPIM "noncommutative stack"
