Is there a universal property for the ultraproduct?
Solution 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 $\left( \prod_{i \in I} X_i \right) / U$.