ct.category theory - Does sheafification preserve sheaves for a different topology?

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"