In this question it was asked what relation, if any, exists between the notions of a monomorphism in a general category $\mathcal C$ and an injective map. Of course the question only makes sense if $\mathcal C$ is a concretizable category. If $\pi:\mathcal C\to Set$ is a faithful functor, we may regard a morphism $f:X\to Y$ of objects of $\mathcal C$ as an injection (relative to $\pi$) if $\pi(f)$ is injective in $Set$.

One then notices that this notion depends very much on the choice of concretization, and thus that this notion is not an intrinsic property of the category itself.

However, Emil Jerabek noticed that it is natural to ask the following: given a concretizable category, does it admit a concretization which preserves monomorphisms?

Apologies if this question is trivial, hopelessly too general, or of no interest to category theorists.