How are the two natural ways to define “the category of models of a first-order theory $T$” related?

Question

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Elem{Elem}$Background/Motivation: Inspired by an interesting question by Joel, I’ve been wondering about the relationship between two very natural ways to define the category of “all models of $T$” where $T$ is a first-order theory.

Let us assume that $T$ is a complete theory with infinite models. Then on the one hand we can define the category $\Mod(T)$ whose objects are all the models of $T$ and whose morphisms are all homomorphisms in the sense of model theory – i.e. functions $\varphi\colon M \rightarrow N$ such that

For any $n$-ary relation $R$ in the language of $T$, if $M \models R^M(a_1, \ldots, a_n)$, then $N \models R^N(\varphi(a_1), \ldots, \varphi(a_n))$;

and

$\varphi(f^M(a_1, \ldots, a_n)) = f^N(\varphi(a_1), \ldots, \varphi(a_n))$ for any $n$-ary function symbol $f$ in the language of $T$.

Also, one can define another category $\Elem(T)$ whose objects are also all the models of $T$, but whose morphisms are only the elementary embeddings, that is, functions which preserve the truth of all first-order formulas. As a model theorist, I’m more used to thinking about the category $\Elem(T)$, and this latter category arises naturally if one cares about which sets are definable but one does not particularly care about which sets are definable by positive quantifier-free formulas.

Question: What sorts of category-theoretic properties automatically transfer from $\Mod(T)$ to $\Elem(T)$, or from $\Elem(T)$ to $\Mod(T)$?

To be clear, by a “category-theoretic property” I mean something that is preserved by an equivalence of categories.

Another related question is:

Question: Suppose we have a set of category-theoretic properties which we know characterize all the categories $C$ which are equivalent to $\Mod(T)$ for some $T$ [or to $\Elem(T)$ for some $T$]. Can we use this to characterize the categories which are equivalent to $\Elem(T)$ [respectively, $\Mod(T)$] for some $T$?

Here are a couple of basic facts I know, which may or may not be useful here. First of all, every category $\Elem(T)$ is equivalent to a category $\Mod(T')$ for some other theory $T'$ – namely, the “Morleyization” $T'$ of $T$, where we expand the language by adding new predicates for every definable set (and iterating $\omega$ times), thereby forcing $T'$ to have quantifier elimination. However, it is certainly not true that every category $\Mod(T)$ is equivalent to a category of the form $\Elem(T')$ – for instance, $\Mod(T)$ might not have colimits of $\omega$-directed chains, but $\Elem(T)$ always will (by Tarski’s elementary chain theorem).

Addendum: As Joel pointed out, there is a third possible notion of “morphism” for this category: the “strong homomorphisms” $\varphi\colon M \rightarrow N$ which commute with the interpretations of function symbols and have the property that for any $n$-ary relation $R$ in the language of $T$, $$M \models R^M(a_1, \ldots, a_n) \iff N \models R^N(\varphi(a_1), \ldots, \varphi(a_n)).$$

I’d also be interested to learn about any relationships between the category of models with strong homomorphisms and the other two categories above.

Answer 1

Well, that's not an answer to your question but perhaps it is still a comment you might useful. I happend to have spent some time thinking about how to define a category of models of a theory, and it appeared to me that a reasonable way is to consider the category of families of models of a theory. There are not much indications that this is a good way, though, but still, you can define something like a model category structure on this category, and, perhaps more importantly, this model category structure does give rise to an interesting set-theoretic invariant (the covering number) in the (model theoretically degenerate) case of the theory of equality (and is an actually model category). I am not sure whether you find this helpful, though, and it is definitely not an answer to your question. Actually, I've written a text about this, sort of a report on whatever little I understand now...