I am looking for a way to describe concrete categories with "good properties" in my work, and one of the properties I found that I want a concrete category $C$ to have is that whenever $g$ is a monomorphism in $C$, $f$ is a morphism in Set and $g \circ f$ is a monomorphism in $C$, it follows that $f$ is a morphism in $C$.
Effectively, I want the concrete category $C$ to be such that whenever $X$ is an object in $C$ and $A$ and $B$ are subobjects of $X$ such that $A \subseteq B$ in Set (as subobjects of $X$), then $A \subseteq B$ in $C$ (as subobjects of $X$). In particular, if $A$ and $B$ are subsets of the set $X$ such that $A \subseteq B$ as sets, I want the the inclusion map of $A$ into $B$ to be a morphism in the category $C$.
So, is there a name for such a property of a concrete category? Or perhaps a name for such a property in any subcategory of another category?