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.