Let take a first order theory $T$ in a language $\mathcal L$. One can form the category $\mathrm{Mod}(T)$ whose
- objects are the models of $T$ (those $\mathcal L$-structures $\mathfrak M$ such that $\mathfrak M \models T$),
- arrows are the homomorphism of $\mathcal L$-structures. (One might also want to take only the embedding as arrows to have more elementary-like properties.)
Then, how much does the theory $T$ (or an axiomatisation of $T$) tell about the shape of the category $\mathrm{Mod}(T)$ ?
This question came to me when studying $\mathsf C$-valued (pre)sheafs on a topological space, when $\mathsf C$ was a category among : sets, groups, abelian groups, rings, etc. For all those categories $\mathsf C$, the underlying sets of a stalk of a (pre)sheaf is the stalk of the postcomposite (pre)sheaf with the forgetful functor to sets. This for the one and good reason that the forgetful functor $U \colon \mathsf C \to \mathsf{Set}$ commutes with filtered colimits. The question is :
Could such a property about the forgetful functor be determined from the properties of the theory $T$ ?
To continue with the example of $U$ commuting with filtered colimits, it seems that the theory of groups (respectively abelian groups, rings) admitting a universal axiomatization (in the correct language of course) has something to do with it… (I'm not entirely sure, this insight comes from the special case of a growing union.)
I principally look for references or keywords, as I have no clue about the name of such a field.
P.S. : along my search on the net, I came across the book Accessible Categories: The Foundations of Categorical Model Theory by Makkai and Paré, but it seems kind of hard to apprehend and I can not determine if it could answer my question.