This depends on what definition of *category* you have taken: do you assume the objects form a set?

(I’ll avoid the terminology “small category”, since while this is sometimes used to mean “objects form a set”, it’s also used to mean “objects form a $U$-set”, for some universe $U$ whose sets have been earlier designated as “small”.)

**If you’ve taken the definition where objects of a category are always assumed to form a set, then there’s no problem: just use the axiom of choice.** For each $X, Y$ in $\operatorname{ob} C$, you know there exist some set in $U$ and isomorphism $C(X,Y) \cong U$; so by the axiom of choice, there’s a function which chooses, for each $X$, $Y$, some such set and isomorphism. Given this, the construction of the hom-functor is straightforward.

**If you’re ***not* assuming that the objects of a category form a set — i.e. you define a category to have a class of objects — then life gets rather subtle, because categories in this sense aren’t something you can really talk about inside ZFC. You can represent classes by predicates, and so talk about individual categories (or parametrised families of categories) schematically — so any theorem about arbitrary categories is really a theorem *schema*. Or you can extend your language to something like NBG set theory, where you really can talk about classes (and hence categories) directly.

In either of those setups of “large categories”, **I don’t see how you can define the hom-functor for an arbitrary $U$-category without invoking something approaching the axiom of global choice,** which essentially gives you choice functions on classes instead of sets. Given global choice, we can construct hom-functors essentially by re-running the argument used for the small case above.

Using the language of NBG, one can ask whether the existence of hom-functors for all such categories is (a) independent of NBG (which is conservative over ZFC), or (b) equivalent over NBG to the axiom of global choice. It’s equivalent to the statement “there is a global choice function for non-empty $U$-small sets”, if I’m not mistaken. My guess would be that this statement is independent of NBG, but strictly weaker than full global choice. I can’t substantiate either part of this, off the top of my head, but I think for someone less rusty than me on the relevant techniques, it should be fairly standard.