I am citing the definition of the Hitting Set Problem from (Gardy & Johnson, 1979):
INSTANCE: Collection $C$ of subsets of a set $S$, a positive integer $K$.
QUESTION: Does $S$ contains a hitting set for $C$ of size $K$ or less, that is, a subset $S' \subseteq S$ with $|S'| \leq k$ and such that $S'$ contains at least one element from each subset in $C$?
The problem definitions from other sources (none of which elaborates on the size of $C$ in relation to $|S|$):
Given a set $S$ with $|S| = n$, a collection $C$ of its subsets can have up to $2^n$ elements (where each element is a subset of $S$).
Given a certificate, $S' \subseteq S$, of the Hitting Set Problem, how could we possibly check if $S'$ contains at least one element from each subset in $C$ in polynomial time w.r.t $n$?