I've been reading Abstract and Concrete Categories: The Joy of Cats by Adámek, et al. They take a slightly different approach to defining categories, since they operate with a theory of collections that has three sorts: sets, classes (collections "too big" to be sets), and conglomerates (collections "too big" to be classes).
They don't give a definition of "locally small category" (unless I missed it) but they distinguish between quasicategories and categories. The way they define categories simpliciter seems to coincide with how I've seen locally small categories defined -- a class-sized collection of objects with only hom-sets (no "hom-classes") for the collections of arrows between objects.
A consequence of this (I believe) is that the collection of morphisms of a category will be at most class-sized. This comports with the usual two sorted collection approaches to category theory that I've seen since they don't have any larger-than-class-sized notion of collection so they wouldn't be able to make sense of anything larger than a class-sized collection of morphisms anyway.
This got me to wondering what the nice-making properties of local smallness really were. A few questions on this front:
Is it crucial that the collection of morphisms be at most class-sized? Or is it enough that you have only hom-sets?
Would a quasicategory with a conglomerate sized collection of objects but only hom-sets have the nice-making features of locally small categories? Obviously their collection of morphisms would be conglomerate-sized because there would be conglomerate-many identity arrows on the objects, but is this something that would render them "less nice" to use than locally small categories (i.e., assuming only class-many objects)?
My speculative thought is that the restriction to class-many objects is necessary because you want something like a functorial embedding of the locally small categories into the category of sets, but since the universe of sets is class-size, this wouldn't be possible if you had conglomerate-many objects (and hence conglomerate-many morphisms). Is something along these lines right?
TL;DR What's so great about locally small categories, and if there were more than class-many objects in a (quasi)category with only hom-sets would that be "just as good" as a locally small category?