ag.algebraic geometry - Hilbert scheme of points

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?