Locally small categories, $\mathcal{U}$-categories and equivalence

Question

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?

Answer 1

No. The trivial category with one object and one morphism is equivalent to a category with arbitrary nonempty set of objects: indeed, take a preorder, where $x\le y$ for every objects $x$ and $y$, then this preorder is equivalent to the trivial category. But the set of objects of this preorder may be an arbitrary nonempty set, hence you can take a set with cardinality bigger than $\mathcal{U}$ as its set of objects, so this preorder will not be a $\mathcal{U}$-category (by your definition of $\mathcal{U}$-category, which doesn't coincide with the standard definition, see my answer on your previous question).