Assume GCH. Suppose $j: V\to M$ is an elementary embedding such that $\mathrm{crit}(j)=\kappa$, ${}^\kappa M \subset M$, $M\supset V_{\kappa+2}$. We can assume $j$ is defined from some extender of length $\kappa^{++}$. Hence, in particular, we know $\kappa^{++}<j(\kappa) < \kappa^{+++}$.

For each $\delta<j(\kappa)$, let $U_\delta$ be an ultrafilter on $\kappa$ be defined such that $A\in U_\delta$ iff $\delta\in j(A)$. Is it true that for any $\delta <\delta' < j(\kappa)$, $U_{\delta}\neq U_{\delta'}$? It is true if $\delta<\kappa^{++}$ since $\mathrm{Ult}(V, U_\delta)$, $[id]_{U_\delta}=\delta$. Of course if the length of the extender is too long then this is not true since the number of ultrafilters on $\kappa$ is $\kappa^{++}$.

Assume GCH. Suppose $j: V\to M$ is an elementary embedding such that $\mathrm{crit}(j)=\kappa$, ${}^\kappa M \subset M$, $M\supset V_{\kappa+2}$. We can assume $j$ is defined from some extender of length $\kappa^{++}$. Hence, in particular, we know $\kappa^{++}<j(\kappa) < \kappa^{+++}$.

For each $\delta<j(\kappa)$, let $U_\delta$ be an ultrafilter on $\kappa$ be defined such that $A\in U_\delta$ iff $\delta\in j(A)$. Is it true that for any $\delta <\delta' < j(\kappa)$, $U_{\delta}\neq U_{\delta'}$? It is true if $\delta<\kappa^{++}$ since $\mathrm{Ult}(V, U_\delta)$, $[id]_{U_\delta}=\delta$. Of course if the length of the extender is too long then this is not true since the number of ultrafilters on $\kappa$ is $\kappa^{++}$. In particular with appropriate hypothesis, it is possible that $j=j_E$ where $E$ is some extender on $[\delta]^{<\omega}$ where $\delta\in \kappa^{+++}\cap \mathrm{cf}(\kappa^{++})$ and some $\alpha\neq\alpha'<j(\kappa)$ satisfy that $U_\alpha=U_{\alpha'}$. But $\delta$ may not be $\kappa^{++}$.