Work in ZF. Let $\lambda$ be strongly inaccessible. Let $\kappa$ be such that for every $\alpha\lt\lambda$, there is some $j: V_\lambda\prec V_\lambda$, with critical point $\kappa$, such that $j(\kappa)\gt\alpha$.

In this setting, can we define a sequence $A=\langle j_\alpha: V_\lambda\prec V_\lambda\mid \alpha\lt\lambda\rangle$, such that every $j_\alpha$ is a non-trivial elementary embedding with critical point $\kappa$, and $j_\alpha(\kappa)\gt\alpha$?

(Note: $A$ needs to be a sequence, not just the set of all embeddings.)

**Motivation:** I was doing research on large cardinals without choice, and was attempting to use a similar collection of embedding to $A$ to give a first order definition of cardinals that were $n$-huge.