 Moved critical point is not an ambient cardinal under GCH

For a measure ultrapower $j:V_{\lambda }\to M$ with critical point $\kappa$, there are at most $2^{\kappa }$ equivalence classes of functions $\kappa \to \kappa$, so $|j(\kappa )|\le 2^{\kappa }$ in $V_{\lambda }$. The target $M$ regards $j(\kappa )$ as measurable and contains $V_{\kappa +1}$, so it sees $(2^{\kappa }{)}^M<j(\kappa )$. Under the Generalized continuum hypothesis, this places $j(\kappa )$ strictly above the ambient ${\kappa }^{+}$ while its ambient cardinality is at most ${\kappa }^{+}$; hence it is not an ambient cardinal.