Suppose $j : V \to M$ is $\lambda$-supercompactness embedding derived from an $\kappa$-complete normal ultrafilter $U$ on $P_\kappa(\lambda)$, $\lambda$ regular. Suppose $\eta$ is an ordinal such that $\sup(j[\lambda]) \leq \eta < j(\lambda)$. Let $W$ be the uniform ultrafilter on $\lambda$ defined by $X \in W$ iff $\eta \in j(X)$. Let $i : V \to N$ be the ultrapower embedding by $W$. There is a factor map $k : N \to M$ given by $$k([f]_W) = j(f)(\eta).$$

Question: What is the relationship between $N$ and $M$? What is the critical point of $k$? How closed is $N$?

We can ask similar questions about $\lambda$-strong embeddings.