Work in a foundation that admits a countable hierarchy of notions of 'set', and say that a category is *$n$-small* iff its object and arrow collections are $n$-sets. Denote the category of $n$-sets by ${\bf Set}_n$.

For each $n$-small category $\mathcal{C}$ and each object $X\in{\bf Ob}_\mathcal{C}$, define $${\bf Hom}_\mathcal{C}(\ \ ,X)=cod^{-1}(X)\subseteq{\bf Hom}_\mathcal{C}.$$ $$\Big(=\bigcup_{W\in{\bf Ob}_\mathcal{C}}{\bf Hom}_\mathcal{C}(W,X).\Big)$$ For each $n$-small category $\mathcal{C}$, define a functor $${\bf Hom}_\mathcal{C}(\ \ ,-):\mathcal{C}\to{\bf Set}_{n+1}$$ $$X\mapsto{\bf Hom}_\mathcal{C}(\ \ ,X)$$ $$f:X\to Y\longmapsto f\circ:{\bf Hom}_\mathcal{C}(\ \ ,X)\to{\bf Hom}_\mathcal{C}(\ \ ,Y).$$
${\bf Hom}_\mathcal{C}(\ \ ,-)$ is always faithful, so $n$-small categories are never abstract. Working in an [appropriate foundation](https://arxiv.org/abs/2206.10060) (disclaimer: I am the author of this paper) all classically considered 'abstract' categories like Freyd's homotopy category are just $0$-large $1$-small categories, admitting canonical faithful functors into ${\bf Set}_2$.

Taking this view, it seems like categories are only 'abstract' if we work in a set-theoretical foundation that is 'too small' to see the larger categories of sets that all categories embed into. This makes me wonder,

>Are there any categories we care about that remain abstract in set theories admitting a countable hierarchy of notions of 'set'? What if we extend the hierarchy of notions of set to arbitrary ordinal heights?