Universal family and open and closed Hilbert polynomial strata

Apply the natural representing bijection to the identity morphism of $H_{X\mathrm{/}S}$. Its image is the universal family $\mathcal{Z}$. Naturality identifies the image of every $T\to H$ with its pullback, and bijectivity gives existence and uniqueness of the classifying map. The fixed-polynomial components represent exactly the polynomial subfunctors, so the image of a polynomial-$P$ open and closed locus lands in that component and its inverse image is precisely this locus.

The full functor is the set of embedded flat finitely presented families before any polarization is chosen. Hence two constructions made with different relatively ample bundles represent the identical functor; the natural identification gives unique mutually inverse scheme maps carrying universal families to each other. Its stratum labels can change because their polynomials are computed with the selected bundle. The same representing bijections and the canonical Cartesian identifications in [F1] prove compatibility of these intrinsic identifications, classifying maps, and families with every base change.