Let $S$ be a set. Then the improper subset of $S$ is $S$ itself, viewed as a subset of itself.

As the subsets of $S$ correspond to the predicates on $S$ and to the unary relations on $S$, so the improper subset corresponds to the predicate that is always true and the relation which always holds.

As the subsets of $S$ correspond to the injections to $S$ (up to isomorphism in the slice category$Set/S$), so the improper subset corresponds to the identity function$\id_S\colon S \to S$ (or to any bijection to $S$, since these are all isomorphic in $Set/S$).