ct.category theory – $ mathscr {U} $ – categories and $ mathsf {Hom} $ – functors

Leave $ 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 $ X $ a $ mathscr {U} $-category if for some $ X, Y in mathsf {C} $, $ mathsf {Hom_C} (X, Y) $ is $ mathscr {U} $-little.

Assume $ mathsf {ZFC} $ As our foundational system (not Bourbaki's set theory).

Leave $ mathsf {C} $ be a $ mathscr {U} $-category and leaves $ mathscr {U} text {-} mathsf {Set} $ be a set of all the sets that belong to $ mathscr {U} $.

How do we build a $ mathsf {Hom} $-functor $ mathsf {Hom_C} (X, -) colon mathsf {C} to mathscr {U} text {-} mathsf {Set} $? Note for each $ Y in mathsf {C} $, $ mathsf {Hom_C} (X, Y) $ does not belong to $ mathscr {U} text {-} mathsf {Set} $, but it is isomorphic to a set there. Grothendieck in SGA uses Bourbaki's set theory and $ tau $ operator of choice (also axiom $ mathscr {U} $B), while in $ mathsf {ZFC} $ we do not have that

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