Every language in NP reduces to Hitting Set in polynomial time, by a reduction whose output instances stay of size polynomial in the length of the input and satisfy $2 \le k \le n$.

This is Karp's theorem. The reduction is Cook's theorem, followed by the reduction from satisfiability of $HittingSetFromSat$: a formula is satisfiable exactly when its instance has a hitting set of the required size, and the reduction runs in polynomial time on a Turing machine writing the binary word of the instance. The theorem is stated twice: as a many-one reduction into the language of Hitting Set, and in the form the reduction of this submission consumes it.

The two restrictions on $k$ cost nothing. A hitting set of size exactly $k$ cannot exist once $k$ exceeds the universe, so the upper one is a normalization; for the lower one, pad an instance with one fresh element and the singleton set containing it, which forces that element into every hitting set and raises $k$ by one.

The universe of an emitted instance is no larger than $4 + m$ plus the total size of the sets. The reduction from satisfiability satisfies this, since its pairs alone cover the universe. The clause is what lets the next reduction, which reads the word of the emitted instance, be polynomial-time in that word: the shop it builds has more than $n^2$ jobs, so a universe larger than the sets that present it — which a word of $O(\log n)$ bits could name — could not be written by any machine polynomial in the word. It is the same restriction the admissible words of the fixed-parameter statements carry, that the universe is no larger than the word that presents it.

That the size of the emitted instance stays polynomially bounded is automatic for a polynomial-time reduction, since an instance cannot be larger than what was written. It is stated because it is what makes the composed reduction of this submission a strong NP-hardness statement: the numbers of the constructed shop are polynomial in $n$, $m$ and $k$, hence in the length of the original input.