If $T$ is a complete first-order theory and $\kappa$ is a cardinal, let $\mathrm{Mod}\_\kappa(T)$ be (a skeleton of) the category of $\kappa$-small models of $T$ (i.e. of cardinality $<\kappa$), with elementary embeddings as morphisms. What are the possible cardinalities of (the set of morphisms of) $\mathrm{Mod}\_\kappa(T)$? And more specifically: what are the possible cardinalities of the hom-sets $\mathrm{Hom}\_{\mathrm{Mod}\_\kappa(T)}(M,N)$ for various $M,N \in \mathrm{Mod}\_\kappa(T)$.


From a categorical perspective, this is a natural variant on classification theory and the question of the number of nonisomorphic models, which is about the cardinality of the set of *objects* of $\mathrm{Mod}\_\kappa(T)$. It also provides a variation on Vaught's conjecture to consider.


For example, suppose the language is countable and $\kappa = \aleph\_1$ and there is an infinite model. There's an obvious upper bound on $|\mathrm{Mod}\_\kappa(T)|$ of $2^{\aleph\_0}$, which is attained in all the examples I can think of (but I'm not actually a model theorist!). There's also a topology on the homsets given by pointwise convergence, with respect to which composition is continuous, which is a metric topology in the case of countable models, so it's tempting to try to construct perfect sets of embeddings.



How many elementary embeddings can there be?
It is a fact (following from the Ehrenfeucht–Mostowski theorem) that for every complete theory $T$ and for every $\lambda \geq |T|$, there is $M \models T$ with $|M| = \lambda$ and $M$ having $2^\lambda$-many automorphisms (assuming $T$ has infinite models). So if I'm understanding your question correctly then what you denote as $\mbox{Mod}\_\kappa(T)$ always has cardinality $2^{<\kappa}$, at least for $\kappa > |T|$.


PS: (simplified example) To partially address your second question: for every cardinal $\lambda$ (possibly $\lambda$ finite or $0$) there are $M \equiv N$ structures in a countable language such that there are exactly $\lambda$ elementary embeddings from $M$ to $N$. Let $\mathcal{L}$ be the language with infinitely many constant symbols $(c\_m: m < \omega)$ and let $T$ say that they are all distinct. For $\lambda > 0$, let $M$ be the model with exactly one unsorted element and let $N$ be the model with exactly $\lambda$ unsorted elements; then there are $\lambda$ elementary embeddings from $M$ to $N$. For $\lambda = 0$ let $M$ have 1 unsorted element and let $N$ have no unsorted elements.


PPS: On the other hand if $M, N$ are countable then the possibilities are exactly $0, 1, 2, \ldots, \aleph\_0, 2^{\aleph\_0}$. From the above examples we have seen these are all possible, on the other hand the space of elementary embeddings from $M$ to $N$ is a Polish space (completely metrizable space) so either is countable or has a perfect subset. (The subtleties of Vaught's conjecture are that we are looking at models up to isomorphism, so an equivalence relation on a Polish space.)


