Is this a typo or a misunderstanding? (Fine Structure in the Handbook of Set Theory) I'm reading through Schinder and Zeman's handbook chapter on fine structure, and encountered an issue trying to understand the proof of Lemma 7.4.  The statement of the lemma is as follows

Let $M$ be acceptable, and let $ν\in p\in P_M$.  The following are equivalent.

*

*$W^{ν,p}_M\in M$


*There is a witness $W$ for $ν\in p$ with respect to $M$ and $p$ such that $W\in M$.

The trouble I'm having is understanding the definition of the map $σ^*:W^{ν,p}_M\rightarrow W$ and the use of "$h_N$" in particular.
Explicitly, let $\langle W,r\rangle $ be the witness in (2), and let $σ:W^{ν,p}_M\rightarrow M$ be the inverse of the transitive collapse. $σ^*$ is defined by, for $ξ<ν$,
$$ σ^*\left(h_{W^{ν,p}_M}(ξ, σ^{-1}(p\setminus ν+1))\right)=h_W(ξ,r)\text{.}$$
Now I know that Schinder and Zeman like to use $h_N$ both as a skolem hull and as a skolem function, but I don't know how either of those makes sense here.
If we're using $h_N$ as a skolem function, shouldn't we have an argument involving the $n$th formula, $n<ω$?  If we're using $h_N$ as a skolem hull --- i.e. interpretting $h_W(ξ,r)=h_W(\{ξ,r\})$ --- is it even clear that these hulls are elements of $W^{ν,p}_M$ or $W$, since supposedly $σ^*:W^{ν,p}_M\rightarrow W$?
Really I guess I'm just kind of confused about the proof and notation.
 A: Part of your confusion seems to be caused by a(n) (common) abuse of notation.
Let $N$ be an acceptable (amenable suffices) $\mathcal{J}$-structure. Then there is a uniformly definable $\Sigma_1$-Skolem function for $N$ -- call it $h_N$. I.e. in a fixed (recursive) enumaration $(\phi_n \mid n < \omega)$ of all $\Sigma_0$-formulae in the language of $N$, we have, for all $\vec{p} \in N$
$$
N \models \exists x \phi_n[x,\vec{p}] \iff h_N(n, \vec{p}) \text{ is defined and } N \models \phi_n[h_N(n, \vec{p}), \vec{p}].
$$
Let us now try to understand what's going on with $\sigma^*$:
$$
W^{\nu,p}_M = \mathrm{cHull}_{\Sigma_1}^M(\nu \cup (p \setminus (\nu + 1))
$$
is the transitive collapse of the $\Sigma_1$-hull of $M$ over $\nu \cup (p \setminus (\nu + 1))$. In particular, the Mostowski collapse
$$
\sigma \colon W^{\nu,p}_M \to M
$$
is $\Sigma_1$-elementary, so that $W^{\nu,p}_{M}$ is itself an acceptable $\mathcal{J}$-structure (as this is a $\mathcal{Q}$-property).
Claim. $\bar{p} := \sigma^{-1}(p \setminus (\nu + 1))$ is a very good parameter of $W^{\nu,p}_M$.
Proof. Let $x \in W^{\nu, p}_M$. Then $\sigma(x) \in \mathrm{Hull}_{\Sigma_1}^{M}(\nu \cup (p \setminus (\nu + 1) ) \prec_1 M$. Hence there is some $\xi < \nu$ and some $n < \omega$ such that $\sigma(x) = h_M(n, (\xi, (\xi, \sigma(\bar{p})))$.
By the uniform definability of $h_N$ (using that $\sigma$ is $\Sigma_1$-preserving and $\sigma \restriction \nu = \mathrm{id}$), it follows that $x = h_{W^{\nu, p}_M}(n, (\xi, \bar{p}))$. Q.E.D.
Now define, for all $n < \omega$ and all $\xi < \nu$,
$$
\sigma^* \colon W^{\nu, p}_M \to W, h_{W^{\nu, p}_M}(n, (\xi, \bar{p})) \mapsto h_W(n, (\xi, r)).
$$

It is here that the abuse of notation took place: People often drop the natural number parameter when talking about Skolem functions of $\mathcal{J}$-structures. Probably because they think about the generated hulls and not the actual Skolem functions when writing down these kinds of arguments.

Exercise. $\sigma^*$ is a total, well-defined, $\Sigma_0$-elementary embedding.
(Hint: You need to use that $(W,r)$ is a generalized witness.)
Exercise. If we let $\alpha = \sup h_W(\nu \cup \{r \}) \cap \mathrm{Ord})$, then
$$
\sigma^* \colon W^{\nu, p}_M \to \mathcal{J}^W_{\alpha} (\dagger)
$$
is $\Sigma_0$-elementary and cofinal. Therefore it is $\Sigma_1$-elementary.

$(\dagger)$ For an acceptable $\mathcal{J}$ structure $N$ and $\alpha \le N \cap \mathrm{Ord}$, $\mathcal{J}^N_{\alpha}$ is the $\alpha$-th level of $N$ in its $\mathcal{J}$-hierarchy, i.e. if $N = (J_{\beta}^A; \in, A)$, then $\mathcal{J}^{N}_{\alpha} = (J_{\alpha}^A; \in, A \cap J_{\alpha}^A)$.
