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 $$mathsf {ZFC}$$?