Maybe this is an elementary question.

Suppose that $U$ is a non-principal $\kappa$-complete ultrafilter on $\kappa$ and consider the standard ultrapower $M\cong \textrm{Ult}_U(V)$ along with the corresponding elementary embedding $j_U : V \rightarrow M$,.

We know that if $U$ is in addition normal, then every element $y\in M$ can be written in the form $j(f)(\kappa)$ (more precisely, if $y=[f]_U \in M$ then $y=j(f)(\kappa)$, which follows from the fact that $[id]_U=\kappa$).

Of course, we can always pick $U$ to be normal, but the question is what happens in the case that we don't. More precisely:

Assume that in $M$, $[g]=\kappa<[id]_U$, for some $g:\kappa\rightarrow \kappa$. Does it follow that for every $y\in M$, $y=j(f)(\kappa)$ for some function $f$ on $\kappa$?

My guess is that the desired would follow if can pick the function $g$ that represents $\kappa$ to be 1-1 and such that $\textrm{range}(g)\in U$. Is this true? Can we do this?