I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology.
My question is:
How about the sheafification functor from the category of presheaves of "sets" on $C$ to the category of sheaves of "sets" on $C$?
Is this an exact functor? (i.e. preserving finite limits and finite colimits?)
If so, how can one prove it?
In fact, I want to know whether sheafification functor preserves cartesian products or not.
Please give me any advice.