This question is related to this one Equivalence of some statements regarding Vopenka's principle and some $V_\kappa$ for inaccessible $\kappa$
In particular, on page 336 of Kanamori's book on large cardinal, he wants to show the equivalence that for inaccessible $\kappa$, the Vopenka filter is a proper filter over $\kappa$ iff $V_\kappa$ is a model of VP iff $\kappa$ is Vopenka.
In one step he writes that each sequence of structure $<A_\alpha \ | \ \alpha < \kappa> \in {}^{\kappa}V_\kappa$ can be encoded into a natural sequence $M_\alpha$ so that if $X$ is Vopenka in $\kappa$, there are $\alpha < \beta < \kappa$ and $j:A_\alpha \prec A_\beta$ such that if it has a critical point, it belongs to $X$. For this coding, the natural one and the one suggested by the answer to the above link, is to find, for any $A_\alpha$, some high enough $M_\alpha= <V_{f(\alpha)}, \{\alpha\}, A_\alpha>$, so that for any $j:M_\alpha \prec M_\beta$ we have $j|A_\alpha: A_\alpha \prec A_\beta$ is the desired elementary embedding.
My first question is that why, if $j|A_\alpha$ has a critical point, does the critical point have to be in $X$? The critical point of $j|A_\alpha$ might not be the critical point of $j$ so it might not be in $X$. Also the fact that the critical point is in $X$ does not seem to be required for the derivation of the equivalence mentioned above, it only requires the case where $X=\kappa$.
The second question is that why, in his proof of 24.14, does he have to include the $f|\alpha$ in $N_\alpha$? It does not seem to be used in the proof.
Thank you!