[A previous question on the categorical nature of ultraproducts](https://mathoverflow.net/questions/11261/is-the-ultraproduct-concept-fundamentally-category-theoretic) had great answers, mostly categorically characterizing ultraproducts in the category of $L$-structures and *homomorphisms* for a fixed signature $L$. This is understandable, given one of the most important use of ultraproducts is to give a syntax-free characterization of elementary equivalence.


Here I would like to ask sort of its opposite: can one characterize ultraproducts in the category of $L$-structures and *elementary embeddings*? (This is related to the question of whether or not ultraproducts are a genuinely model-theoretic notion.)


This sounds like what experts of AECs study, but I was not able to find relevant literature.


*Addendum*: Answers focusing on ultra\_powers\_ are also welcome. (In fact, that was my original motivation.)



Ultraproducts in the category of structures and elementary embeddings
Since you asked about ultraproducts, and not ultrapowers, let me argue that the answer must be negative. The reason is that the category of $L$-structures under elementary embeddings is partitioned into disconnected components by the theories of those structures, since elementary embeddings must preserve the theory of the structure and so there can be no morphisms in this category that connect structures with different theories.


But I may form the ultraproduct $\prod M\_i/U$ of structures $M\_i$ with all different theories, in such a way that the ultraproduct itself had a different theory than each of them. So there will be no morphisms at all that relate any of the $M\_i$'s to each other or to the ultraproduct. There will be no diagrams in which the $M\_i$'s participate along with the ultraproduct, except diagrams that are disconnected into components in the same way as the category is by the theories of the models. For example, there can be no universal property in this category relating the $M\_i$'s to the ultraproduct or to models connected by morphisms to the ultrapower or each other, since there are no morphisms at all relating these structures to any other structures in common—they simply live on different islands. In these circumstances, it seems impossible to find a category-theoretic characterization of the ultraproduct in the category of elementary embeddings.


