Recall that a $J$-structure is an amenable structure of the form ($J_{\alpha}^A,B$) where $A$ and $B$ are predicates and $\alpha$ is a limit ordinal. Then if you let $M=J_{\alpha}^A$, there is a surjective function $f:[\alpha]^{\lneq \omega} \rightarrow M$ which is $\Sigma_1^M$. Can we prove that there exists surjective functions $f:[\alpha]^{\lneq \omega} \rightarrow M$ which are $\Sigma_n^M$ for all $n$<$\omega$?

The proof that there is a surjective function $f:[\alpha]^{\lneq \omega} \rightarrow M$ which is $\Sigma_1^M$ uses that $h_M(\alpha)$ is a $\Sigma_1$ elementary substructure of $M$ ($h_M(\alpha)$ the Skolem hull of $\alpha$)