$\newcommand\Set{\mathbf{Set}}\newcommand\Ob{\mathbf{Ob}}\newcommand\Hom{\mathbf{Hom}}$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 $\Set_n$.

For each $n$-small category $\mathcal{C}$ and each object $X\in\Ob_\mathcal{C}$, define 
\begin{gather*}
\Hom_\mathcal{C}(\ \ ,X)=\operatorname{cod}^{-1}(X)\subseteq\Hom_\mathcal{C} \\
\Bigl({}=\bigcup_{W\in\Ob_\mathcal{C}}\Hom_\mathcal{C}(W,X)\Bigr).
\end{gather*}
For each $n$-small category $\mathcal{C}$, define a functor
\begin{gather*}
\Hom_\mathcal{C}(\ \ ,-):\mathcal{C}\to\Set_{n+1} \\
X\mapsto\Hom_\mathcal{C}(\ \ ,X) \\
f:X\to Y\longmapsto {f\circ{}}:\Hom_\mathcal{C}(\ \ ,X)\to\Hom_\mathcal{C}(\ \ ,Y).
\end{gather*}
$\Hom_\mathcal{C}(\ \ ,-)$ is always faithful, so $n$-small categories are never abstract. Working in an appropriate foundation (see [An axiomatic approach to higher order set theory](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 $\Set_2$.

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?