Reduced schemes from schemes

Question

My homework problem is to prove that the embedding of the category of reduced schemes into the category of all schemes has a right adjoint. The most natural thing to do is to set $O'_x(U) : = O_x(U)/\sqrt{0}$ for any $U \subset (X, O_x)$. I can prove that this assignment is a presheaf but I can't prove that it is a sheaf.