Let $\kappa$ be a measurable cardinal (or other large cardinal) and let $j:V\longrightarrow M$ be a witness. We know that $j(\kappa)$ has large cardinal properties in $M$, but what about $j(\kappa)$ in $V$?


Let me give a couple tentative nonstandard definitions:


* Call a measurable cardinal $\kappa$ *weakly compact preserving*, if
there is an elementary embedding $j$ witnessing measurability of $\kappa$, such that $j(\kappa)$ is weakly compact in $V$.
* Call a measurable cardinal $\kappa$ *reflecting preserving* if $V\_\kappa\prec V\_{j(\kappa)}$, again in $V$.


What I am really interested in is a weakly compact preserving–reflecting preserving measurable cardinal $\kappa$ such that there is a weakly compact cardinal $\lambda>\kappa$ such that $V\_\lambda\prec V\_{j(\lambda)}$.



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> Question: What is the place of such a cardinal in the large cardinals hierarchy?
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Large cardinal properties of $j(\kappa)$
If $\kappa$ is measurable and there is a weakly compact cardinal $\lambda$ above, then there is an elementary embedding $j:V\to M$ with critical point $\kappa$ and $j(\kappa)=\lambda$. The reason is that one may simply iterate a normal measure, which pushes $j(\kappa)$ higher, until it hits that $\lambda$. The same argument works with any kind of large cardinal $\lambda$. But the embedding $j$ is not an ultrapower embedding. Of course, with an ultrapower embedding by an ultrafilter on $\kappa$, the value of $j(\kappa)$ is never a cardinal in $V$, since it is strictly between $2^\kappa$ and its successor.


For your second question, if $V\_\kappa\prec V\_\lambda$ for a class club of $\lambda$, then again we can find $j:V\to M$ with $j(\kappa)$ being one of those $\lambda$, by iterating. The consistency strength of that situation is strictly higher than a proper class of measurable cardinals, since $V\_\kappa$ itself must be a model of that theory. But it is less than a stationary proper class of measurable cardinals, since from that assumption one can find a model with a measurable cardinal $\kappa$ with $V\_\kappa\prec V$ and indeed a stationary class of measurable $\lambda$ above which form an elementary chain $V\_\kappa\prec V\_\lambda$. 


