My question is about the proof of the following proposition from Kanamori's "The Higher Infinite":
Proposition. If $\kappa$ is supercompact and $1$-extendible, then there is a normal ultrafilter over $\kappa$ such that $$\{\alpha \lt \kappa : \alpha \text{ is supercompact}\} \in U.$$ In particular the least supercompact cardinal is not $1$-extendible.
The following part of the proof is the part which I have trouble understanding:
Let $j : V_{\kappa+1} \prec V_{j(\kappa)+1}$ witness the $1$-extendibility of $\kappa$. Since $j(\kappa)$ is inaccessible, $$V_{j(\kappa)+1} \models \kappa \text{ is } \gamma\text{-supercompact for } \kappa \le \gamma \lt j(\kappa).$$
What I have trouble understanding is the following:
I know that $\gamma$-supercompactness can be characterized in terms of the existence of normal ultrafilters over $P_\kappa\gamma$ and for this we only need $j(\kappa)$ to be a limit ordinal for $V_{j(\kappa)+1}$ to contain these ultrafilters. Why do we need the inaccessibility of $j(\kappa)$?
This makes me think I'm missing some obvious point.