This is more or less a followup of [this](https://mathoverflow.net/questions/21667/are-there-any-homotopical-spaces) 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](https://amathew.wordpress.com/2012/01/26/homotopy-is-not-concrete/) 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.



Can one make a category concrete by "enlarging the universe"?
By the Yoneda embedding, any category is a full subcategory of its category of presheaves. For this to be valid, the collection of objects of the category has to be a set. So if you enlarge the universe to make this the case then you can make any category concrete.


That being said, I "don't believe in" (the usefulness of) any of this terminology, either the vocabulary associated with "sets" or that of "concrete" categories.


