Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept.

My question is whether the ultraproduct construction is fundamentally a category-theoretic concept.

The ultraproduct/ultrapower construction of Łos is used pervasively in logic, particularly in model theory and also in set theory, where nearly all of the larger large cardinal axioms can be formulated in terms of the existence of certain kinds of ultrapowers of the universe.

My question is, is the ultraproduct fundamentally a category-theoretic construction, in the sense that it is characterized by some natural category-theoretic universal property? How about the special case of ultrapowers?

I would be very interested, if there were a natural universal characterization in terms of the usual Hom sets for these first order structures, namely, first order elementary embeddings and/or homomorphisms. (Needless to say, I would be much less interested in a characterization that amounted merely to a translation of the Łos construction or of Łos's theorem into category-theoretic language.)

**Background.** Suppose we have a collection of structures M_{i} for i in J, all of the same first order type (e.g. groups, partial orders, graphs, fields, whatever), and U is an ultrafilter on the index set J. This means that U is a nonempty collection of nonempty subsets of J, containing every set or its complement, and closed under intersection and superset. The ultraproduct ΠM_{i} /U consists of equivalence classes [f]_{U}, where f is a function with domain J, with f(i) in M_{i}, and f ∼_{U}g iff {i in J | f(i)=g(i)} in U. One imposes structure on the ultraproduct by saying that a relation holds in the product, if it holds on a set in U, and similarly for functions. Łos's theorem then states that the ultraproduct satisfies a first order formula φ([f]_{u}) if and only if {i in J | M_{i} satisfies φ(f(i))} is in U. That is, truth in the ultraproduct amounts to truth on a U-large set of coordinates. The special case when all M_{i} are the same model M, we arrive at the *ultrapower* M^{J} /U. In this case, there is a natural map from M into M^{J} /U, defined by x maps to [c_{x}]_{U}, where c_{x} is the constant function with value x. It is easy to see that this map is an elementary embedding from M into the ultrapower.

This question is a more focused instance of a probably-too-general question I asked here, and I may have several more in the future.