This is based on a few previous questions.
Can one characterize ultrapowers in the category of L-structures (modeling a fixed complete theory, say) and elementary embeddings?
Previous posts showed that you can do this for ultraproducts in the category of structures and homomorphisms, but not generally for ultraproducts in the other category.
Ultraproducts cannot be characterized in the category of models and homomorphisms (see R. Börger, Remarks on categorical notions of ultraproducts, Abstracts, Sussex Category Meeting 1982). His example adds to the theory of fields a 0-ary relation symbol $R$ which is interpreted as true or false. Models are either $A^+$ or $A^-$ where $A$ is a field and $R$ is interpreted in $A^+$ as true and in $A^-$ as false. Let $F$ be the functor on models sending $A^+$ to $A^+$ and $A^-$ to $A^-$ if $A$ has the characteristic $\neq 0$ and $A^+$ to $A^-$ and $A^-$ to $A^+$ otherwise. Then $F$ is an isomorphism on the category of models. Let $\mathcal U$ be a non-principal ultrafilter on the set of primes and $A=\prod\limits_\mathcal U A_p$ where $A_p$ is a field of characteristic $p$. Then $\prod\limits_\mathcal U A_p^+=A^+$, $F(A^+)=A^-$ and $\prod\limits_\mathcal U F(A_p^+)=A^+$. Thus $F$ does not preserve ultraproducts, hence ultraproduts are not definable in the category of models and homomorphisms.
This example also shows that ultraproducts cannot be characterized on the category of models and elementary embeddings. It suffices to take algebraically closed fields instead of fields where homomorphisms coincide with elementary embeddings.
