I need example that associated sheaf functor doesn't preserve arbitrary products. I think that one can provide an example for sheaves over topological space. Thanks for your help.

An example where sheafification does not preserve arbitrary products is where we take sheaves over a (sober) space $X$ that is not locally connected, for example the space of irrationals or Cantor space. Recall that a Grothendieck topos $E$ is locally connected if the (essentially unique) geometric morphism $\Gamma = f_\ast: E \to Set$ has a left adjoint $f^\ast$ that in turn has a left adjoint. More generally, a geometric morphism $f_\ast: E \to F$ between toposes is an essential geometric morphism if its left adjoint $f^\ast$ has a left adjoint. We have the following facts:



Let $X$ be a space. An abelian group (or a set, if you prefer) $G$ determines a constant presheaf, call it $C_G$, and the associated sheaf is the sheaf of locally constant maps into $G$. Given a family $G_i$ with product $G$, the product of the presheaves $C_{G_i}$ is the presheaf $C_G$, but the product of the associated sheaves is not in general the sheaf of locally constant maps to $G$. (If $X$ is not locally connected then one can easily have a map to $G$ which projects to a locally constant map to $G_i$ for each $i$ but which is not locally constant itself. ) This is presumably a downtoearth special case of what Todd Trimble is saying. 

