Is there a universal property for the ultraproduct?

Question

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?

Answer 1

Let $\mathcal{U}$ be the ultrafilter $U$ considered as a partially ordered set in its own right, and consider the diagram of shape $\mathcal{U}^\mathrm{op}$ where the value at an element $S$ is the product $\prod_{i \in S} X_i$ and the transition maps are the obvious projections. The colimit of this diagram (which is a directed system!) is then the ultraproduct $\left( \prod_{i \in I} X_i \right) / U$.