Category of presheaves

Question

Let $A$ be a category, $\hat A$ be the category of presheaves over $A$. Then how do we know ${\rm Hom}_{\hat A}(X,Y)$ is small enough to be a set? For presheaves $X,Y$ (functors), morphisms (natural transformations) between $X$ and $Y$ form a set?

Answer 1

One interesting fact is that a category $A$ is essentially small if and only if both the categories $A$ and $\hat A$ are locally small.

So if $A$ is small then you're fine, but most categories with their naive definitions (like sets, rings, modules, vector spaces, etc) are large locally small categories, which implies that their presheaf categories are not locally small, so $\mathrm{Hom}_{\hat A}(X, Y)$ can indeed be large.

To deal with this some authors use Grothendieck universes, some authors use set hierarchies (which amounts to the same thing), and some authors just ignore the issue all together.