Let $I$ be a normal ideal on $P_{\kappa} (\lambda)$. Let $V$ denote our ground model. Now we force with the $I$-positive sets, and if $G$ is the resulting generic filter, it can be shown that $G$ is a $V$-ultrafilter on $P_{\kappa} (\lambda)$ extending the dual filter of $I$, which is normal if $I$ is normal.

We can build now the so called generic ultrapower, i.e. we construct inside $V[G]$, $Ult_{G} (V)$ the class of all functions in $V$ with domain $\kappa$ and the usual binary relation $\in^{\ast}$. We assume that the generic ultrapower is well-founded, so we identify it with its transitive collaps $M \cong Ult_{G} (V)$

My question now is: Is this $M$ closed under sequences from $V[G]$ of length $\lambda$, i.e. $M^{\lambda} \cap V[G] = M^{\lambda} \cap M$?

$M$ is closed under sequences from $V$ of length $\lambda$, this is clear to me but I don't have a good argument for the sequences of $V[G]$.