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 (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?

$\newcommand\Set{\mathbf{Set}}\newcommand\Ob{\mathbf{Ob}}\newcommand\Hom{\mathbf{Hom}}$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 (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?