Emily Riehl in her book "Category Theory in Context" states (p.35) that a category equivalent to a locally small category is again locally small.
There are various ways to deal with size issues in category theory. Two most popular ones are theory of classes (such as NBG or MK set theories) and Grothendieck universes. Riehl adopts the first approach, hence a locally small category for her is a category where all $\mathsf{Hom}$-classes are locally small.
A set $\mathcal{U}$ is a Grothendieck universe if
$\forall x \in \mathcal{U}, x \subseteq \mathcal{U}$,
$\forall x,y \in \mathcal{U}, \{x,y\} \in \mathcal{U}$,
$\forall x \in \mathcal{U}, \mathcal{P}(x) \in \mathcal{U}$,
for any family $(x_i)_{i \in I}$ such that $I \in \mathcal{U}$ and $\forall i \in I, x_i \in \mathcal{U}$ we have $\bigcup_{i \in I} x_i \in \mathcal{U}$,
$\mathbb{N} \in \mathcal{U}$.
An analogous concept to a locally small category in theory of universes is a $\mathcal{U}$-category.
Let $\mathcal{U}$ be a universe. A category $\mathsf{C}$ is a $\mathcal{U}$-category if
$\mathsf{Ob(C)} \subseteq \mathcal{U}$,
$\forall X,Y \in \mathsf{Ob(C)}$, $\mathsf{Hom_{C}}(X,Y) \in \mathcal{U}$.
Here subsets of a universe serve as classes and elements of a universe serve as sets.
What I want to know is whether a statement analogous to "any category equivalent to a locally small category is again locally" small holds with respect to $\mathcal{U}$-categories:
Let $\mathcal{U}$ be a universe. Let $\mathsf{C}$ and $\mathsf{D}$ be categories where $\mathsf{D}$ is a $\mathcal{U}$-category. Is it true that if $\mathsf{C}$ and $\mathsf{D}$ are equivalent, then $\mathsf{C}$ is also a $\mathcal{U}$-category?