Consider the category of (flat) families of degree $d$ hypersurfaces in $\mathbb P^n$, i.e. objects are $X \to Y$ whose fibers are degree $d$ hypersurfaces in $\mathbb P^n$, and morphisms are the pullback diagrams. Then, is there a terminal (universal) element in this category? i.e. any family can be uniquely pullbacked by this family.
I have considered the natural family on $|\mathcal O_{\mathbb P^n}(d)|$, but the uniqueness does not hold. I tend to believe the universal family does not exists, but I don't know how to prove it.