Is there a universal property for the ultraproduct?

Given an ultrafilter U on a set I and a collection of ser X_i ($I \in I$) one defines the ultraproduct as the quotient of $\prod X_i $ by the identification $x_i=y_i :\leftrightarrow \{i:x_i=y_i\} \in U$. Is there a possibilityto give this definition in a categorical way by referring to a universal property of a category?

Solutions Collecting From Web of "Is there a universal property for the ultraproduct?"