nLab underlying set

Idea

Given a concrete category, that is a category $C$ equipped with a functor $U$ from $C$ to the category of sets (satisfying certain conditions), we call $U$ a forgetful functor and call $U(x)$, for $x$ an object of $C$, the underlying set of $x$.

In the case where $C$ is explicitly a category of structured sets, then every object $x$ of $C$ is a set ${|x|}$ equipped with some extra structure. In that case, the underlying set of $x$ is precisely this set ${|x|}$.

More generally, if $C$ is concrete over another category $D$, in that we have a faithful functor $U \colon C \to D$, then given any object $x\colon C$, its underlying object of $D$ is $U(x)$.