Let $\mathcal{C}$ be a concrete category, i.e., a category which admits a faithful functor $C:\mathcal{C}\rightarrow \mathsf{Set}$.
It is certainly not the case that $f$ a mono in $\mathcal{C}$ implies that $C(f)$ is a mono (i.e. injective) in $\mathsf{Set}$. However, given $\mathcal{C}$ concrete, is there some concretization $C:\mathcal{C}\rightarrow \mathsf{Set}$ such that $C(f)$ is injective for every mono $f$? If this were the case, to what degree would such a concretization be unique?
The example which I have in mind is the category with two objects $A,B$ and one non-trivial arrow $f:A\rightarrow B$. For any $m,n\ \mathbb{N}$, any choice of $m$ element set $C(A)$, any choice of $n$ element set $C(B)$, and any choice of map $C(f):C(A)\rightarrow C(B)$, $C$ will be a concretization. It is clear that in general $C(f)$ will fail to be injective; however, it is also clear that there are many choices for which it will be. This suggests that the existence result might be true, but that in general such a choice of $C$ will be highly non-unique.
The dual question for epis I don't even bother asking because this fails even for canonical concretizations (e.g. the inclusion $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is an epi in $\mathsf{Ring}$).