Let $\mathcal{U}$ be a universe. The adaptation of the concept of a locally small category to universes is a $\mathcal{U}$-category.

There are two definitions of $\mathcal{U}$ category I've met.

$(1)$ A category $\mathsf{C}$ is a $\mathcal{U}$-category if $\forall X,Y \in \mathsf{C}, \mathsf{Hom_C}(X,Y) \in \mathcal{U}$.

$(2)$ A category $\mathsf{C}$ is a $\mathcal{U}$-category if

$\mathsf{Ob(C)} \subseteq \mathcal{U}$,

$\forall X,Y \in \mathsf{C}, \mathsf{Hom_C}(X,Y) \in \mathcal{U}$.

(SGA takes a different approach because they define a $\mathcal{U}$-small set not as an element of $\mathcal{U}$, but as a set which is equinumerous to some element of $\mathcal{U}$, but this is not something I would like to discuss here)

As you can see, these two definitions differ in whether we require the set of objects of a $\mathcal{U}$-category to be a subset of $\mathcal{U}$. If I'm not mistaken, the straightforward adaptation of the concept of a locally small category seems to be the second approach: with universes, we treat elements of a universe $\mathcal{U}$ as "sets" and subsets of a universes $\mathcal{U}$ as "classes".

What I want to know is which definition is more useful? In particular, is there any reason to restrict the definition of a $\mathcal{U}$-category by requiring the set of object to be a subset of $\mathcal{U}$? It appears to create some problems with functor categories (for example, given a $\mathcal{U}$-small category $\mathsf{C}$ and a $\mathcal{U}$-category $\mathsf{D}$, $[\mathsf{C,D}]$ is not a $\mathcal{U}$-category).