Let $f$ be a Hilbert polynomial, and $X := Hilb_h(P^d_{F_p})$ a Hilbert scheme defined over $F_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, so $X = P^d$.

Which of the following is a sensical question?

What is the family over $X$ induced by pulling back the universal family along $F$? Is there a reasonable way to think about it, that makes it clear that it's not just again the universal family?

If that family is just the universal family again, then in what sense is the Hilbert scheme universal? (As it seems I've obtained the same family from two different maps, which I thought wasn't supposed to happen.)