@AlessandroCodenotti Aren't "Skolem functions" just function-symbols introduced to replace existentials while preserving satisfiability?

The thing I'm trying to understand is the following: Suppose we have $j:V_\delta\prec V_\delta$ an elementary embedding for limit $\delta$. We define $j^+:V_{\delta+1}\to V_{\delta+1}$ by setting, for every $R\subseteq V_\delta$, $j^+(R)=\bigcup_{\alpha<\delta}j(R\cap V_\alpha)$. $j^+$ is not necessarily elementary, but it is the only possibility for an elementary extension of $j$.

Exercise 24.5 in Kanamori's asks to prove that if $j:V_\delta\prec V_\delta$ and $k:V_\delta\prec V_\delta$ are two elementary embeddings then so is $j^+(k)$

And the hint given to show elementarity of $j^+(k)$ is "replace quantifiers by Skolem functions and use a reflection argument"