Let X be a smooth projective variety. We consider the Hilbert scheme X_[n] of points on X. Denote Z the uiversal subscheme in X×X_[n]. We know that Z|(X×ζ)=ζ, where ζ belongs to X[n]. But does the ideal sheaf I_Z have the same universal property, i.e. I_Z|_(X×ζ)=I_ζ?
Yes. The point is that the structure sheaf of $Z$ is flat over $X[n]$, so the formation of its sheaf of ideals commutes with base change on $X[n]$.