Let $\mathscr{U}$ be a universe. Call a set $X$ $\mathscr{U}$-small if there is a set $Y \in \mathscr{U}$ so that $X \cong Y$. Call a category $\mathsf{C}$ a $\mathscr{U}$-category if for any $X,Y \in \mathsf{C}$, $\mathsf{Hom_C}(X,Y)$ is $\mathscr{U}$-small.

Assume $\mathsf{ZFC}$ as our foundational system (not Bourbaki set theory).

Let $\mathsf{C}$ be a $\mathscr{U}$-category and let $\mathscr{U}\text{-}\mathsf{Set}$ be a set of all sets which belong to $\mathscr{U}$.

How do we construct a $\mathsf{Hom}$-functor $\mathsf{Hom_C}(X,-)\colon\mathsf{C}\to\mathscr{U}\text{-}\mathsf{Set}$? Note for every $Y \in \mathsf{C}$, $\mathsf{Hom_C}(X,Y)$ doesn't belong to $\mathscr{U}\text{-}\mathsf{Set}$, but rather is isomorphic to a set in there. Grothendieck in SGA uses Bourbaki set theory and $\tau$ choice operator (also axiom $\mathscr{U}$B), while in $\mathsf{ZFC}$ we don't have that.

Is it even possible to work with these definition in $\mathsf{ZFC}$?