This is more or less a followup of this question. There, it was established that (it is well known that) the homotopy category of topological spaces is not concrete, in other words, there is no faithful functor

$\bf{HoTop}$ $\to$ $\bf{Set}$ .

In this blog post, Akhil Mathew explains that this lack of "concreteness" is due, more or less, to the objects of HoTop having a *proper class* of subobjects (or of quotients).

But... what if one "does not believe" in proper classes? If I'm not mistaken, every "set vs proper class" phenomenon can be seen, in a suitable foundation that assumes the existence of Grothendieck universes, as a "small universe vs large universe" phenomenon. The non-concreteness of HoTop, translated in this language, would mean that there is no faithful functor

$\bf{Ho(U}$-$\bf{Top)}$ $\to$ $\bf{U}$-$\bf{Set}$

where $\bf{U}$ is a Grothendieck universe and $\bf{U}$-$\bf{Set}$ (resp. $\bf{U}$-$\bf{Top}$) is the category of $\bf{U}$-small sets (resp. $\bf{U}$-small topological spaces). Now, what can happen if $\bf{U}\in\bf{V}$ for $\bf{V}$ a larger Grothendieck universe? My question is

Are there faithful functors $\bf{Ho(U}$-$\bf{Top)}$ $\to$ $\bf{V}$-$\bf{Set}$ ?

The same question, of course, can be asked for any other non-concrete category.