Suppose that $\kappa$ is the critical point of $j\colon V\to M$, and suppose that $\mathcal F=(F_\alpha\mid\alpha\leq\kappa)$ is a sequence such that for every limit ordinal $\alpha$, $F_\alpha$ is a sequence of length $\omega_{\alpha+1}$, and $F_\alpha(\beta)$ is a function from $\omega_\alpha\to\omega_\alpha$.

By elementarity, $j(\mathcal F)$ is a sequence of length $j(\kappa)+1$ so we can write it as $(G_\alpha\mid\alpha\leq j(\kappa))$, and clearly $G_\alpha=F_\alpha$ for all $\alpha<\kappa$.

What can we say about $G_\alpha$ and $F_\alpha$? Both are of the same length, since $V$ and $M$ agree on $\kappa^+$, but is it the case that $G_\alpha(\beta)=F_\alpha(\beta)$?

Would anything change if:

1. $\cal F$ is definable canonically? E.g. $V$ is some canonical core model and $\cal F$ is the least sequence satisfying some property.
2. $j$ is an ultrapower embedding? or it isn't an ultrapower embedding?
3. we added so vague coherence property between the different $F_\alpha$s?

Suppose that $\kappa$ is the critical point of $j\colon V\to M$, and suppose that $\mathcal F=(F_\alpha\mid\alpha\leq\kappa)$ is a sequence such that for every limit ordinal $\alpha$, $F_\alpha$ is a sequence of length $\omega_{\alpha+1}$, and $F_\alpha(\beta)$ is a function from $\omega_\alpha\to\omega_\alpha$.

By elementarity, $j(\mathcal F)$ is a sequence of length $j(\kappa)+1$ so we can write it as $(G_\alpha\mid\alpha\leq j(\kappa))$, and clearly $G_\alpha=F_\alpha$ for all $\alpha<\kappa$.

What can we say about $G_\kappa$ and $F_\kappa$? Both are of the same length, since $V$ and $M$ agree on $\kappa^+$, but is it the case that $G_\kappa(\beta)=F_\kappa(\beta)$?

Would anything change if:

1. $\cal F$ is definable canonically? E.g. $V$ is some canonical core model and $\cal F$ is the least sequence satisfying some property.
2. $j$ is an ultrapower embedding? or it isn't an ultrapower embedding?
3. we added so vague coherence property between the different $F_\alpha$s?