Suppose $\kappa$ is $\kappa^{+\omega}$-supercompact. If $U$ is a normal measure on $\mathcal P\_\kappa(\kappa^{+\omega})$, and $j : V \to M$ is the derived embedding, then it is easy to see that $j[\mathcal P\_\kappa(\kappa^{+\omega})] \in M$, and thus $M$ is closed under $\kappa^{+\omega+1}$-sequences by the cardinal arithmetic. We can then derive a normal measure $U'$ on $\mathcal P\_\kappa(\kappa^{+\omega+1})$ by putting $X \in U'$ iff $j[\kappa^{+\omega+1}] \in j(X)$. Let $i : V \to N$ be the ultrapower embedding by $U'$. There is an embedding $k : N \to M$ such that $j = k \circ i$. Define $k([f]) = j(f)(j[\kappa^{+\omega+1}])$.


**Question 1:** Is $k$ the identity?


Now let $\pi : \mathcal P\_\kappa(\kappa^{+\omega+1}) \to \mathcal P\_\kappa(\kappa^{+\omega})$ be $\pi(z) = z \cap \kappa^{+\omega}$. Let $U'' = \{ X \subseteq \mathcal P\_\kappa(\kappa^{+\omega}) : \pi^{-1}[X] \in U' \}$.


**Question 2:** Does $U'' = U$?



supercompactness measure projections
I think about this a little more abstractly using what I call seed theory. See for example my article,  

*Hamkins, Joel David*, [**Canonical seeds and Prikry trees**](http://dx.doi.org/10.2307/2275538), J. Symb. Log. 62, No.2, 373-396 (1997). [ZBL0890.03024](https://zbmath.org/?q=an:0890.03024), which provides an elementary development of this theory. I tried hard in that paper to find a nice, careful treatment of the ideas, which I view as providing a useful framework for answering all these kinds of questions; so I recommend taking a look at the paper, particularly sections 1, 2 and 3. (The paper was adapted from a chapter in my dissertation.)


If you have an elementary embedding $j:V\to M$, then the seed hull of an object $a\in M$ is the class $X\_a=\{j(f)(a)\mid f\in V\}$, and this is an elementary substructure of $M$, which is isomorphic to the ultrapower by the measure $\mu\_a=\{X\mid a\in j(X)\}$ by the association $[f]\_{\mu\_a}\mapsto j(f)(a)$. It is a basic fact that if two seeds $a$ and $b$ generate each other, then the measures $\mu\_a$ and $\mu\_b$ are isomorphic. In particular, any seed $a$ generating all of $M$ will have $j\_{\mu\_a}=j$. 


In your case, for a $\lambda$-supercompactness ultrapower, the seed $j"\lambda$ and the seed $j"\lambda^{<\kappa}$ easily generate each other, for the reasons you gave in your post. Namely, from $j"\lambda$ we can easily construct $j"P\_\kappa(\lambda)$, and from $j"\lambda^{<\kappa}$ we can project to $j"\lambda$. It follows that these seeds generate each other and therefore all of $M$, and so the measures they give are isomorphic, the ultrapowers are in each case $j$ itself and the factor embedding $k$ is the identity. The same is true whenever we have two elements $a$ and $b$ in $M$ that can define each other in $M$ using parameters in $\text{ran}(j)$; the corresponding measures $\mu\_a$ and $\mu\_b$ will be isomorphic and the factor embeddings identical.


