If $\mathcal F$ is a sheaf, then is $\mathcal F (- \times X)$ a sheaf? Let $\mathcal F$ be a sheaf of sets on a site.  Fix an object $X$ of the underlying category of the site, which is assumed to contain a final object and have products.  Define a presheaf $\mathcal G$ by setting $\mathcal G(U) := \mathcal F(U \times X)$.  Is $\mathcal G$ a sheaf?

If $(U_i)$ cover $U$, then $\mathcal{G}(U)$ is supposed to be a certain limit of a diagram of the $\mathcal{G}(U_i)$ and $\mathcal{G}(U_i \times_U U_j)$.  The $U_i \times X$ also cover $U \times X$ (this is the base change axiom for the site), and $(U_i \times_U U_j) \times X = (U_i \times X) \times_{U \times X} (U_j \times X)$, so the analogous diagram for $\mathcal{F}(U \times X)$ is indeed a limit diagram, by the sheaf condition on $\mathcal{F}$.  Thus the answer is yes.
