A concrete category is a category $C$ endowed with a faithful functor $U:C\rightarrow Set$. And if $a$ is an object in $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with codomain $a$.
I’m wondering if subobjects can be related to subsets using $U$. My question is, given a subobject of $a$, does there always exist a monomorphism $f$ in the subobject such that $U(f)$ is an inclusion map from a subset of $U(a)$ to $U(a)$?
If not, does anyone know of a counterexample?