So I started reading the chapter on Indiscrenibles and $0^\#$ in The Higher Infinite and I have some questions on Canonical Skolem terms. I will type the paragraph from the book below, sorry if it's long, but it contains the required definitions.
By 3.3(a) there is a formula $\varphi_0(v_0, v_1)$ that defines in $L$ a well-ordering $\lt_L$ of $L$ s.t. for any limit $\delta \gt \omega$ and $x, y \in L_\delta$, $x \lt_L y$ iff $L_\delta \models \varphi_0[x, y]$.
For each formula $\varphi(v_0, \dots, v_m)$ of $\mathfrak{L}_\in$, define the Canonical Skolem term $t_\varphi$ for $\varphi$ using $\varphi_0$ as follows: $$ t_\varphi(v_1, \dots, v_m) = v_0 \text{ iff }(\forall v_{m+2} \neg \varphi(v_{m+2}, v_1, \dots, v_m) \land v_0 = \emptyset) \lor (\varphi(v_0, v_1, \dots, v_m) \land\forall v_{m+1}( \varphi_0(v_{m+1}, v_0) \rightarrow \neg \varphi(v_{m+1}, v_1, \dots, v_m))). $$
For any $\mathfrak{M} = \langle M, E \rangle$ satisfying the requisite well-ordering properties of $\varphi_0$, the corresponding expansion $\langle M, E, t_{\varphi}^{\mathfrak{M}} \rangle_{\varphi}$, can be considered where the well-defined interpretation $t_{\varphi}^{\mathfrak{M}}$ is a Skolem function for $\varphi$ s.t. $t_{\varphi}^{\mathfrak{M}}$ is the least $y$ according to $\varphi_0$ satisfying $\mathfrak{M} \models \varphi[y, x_1, \dots, x_m]$ when one exists. Note that $\{ t_{\varphi}^{\mathfrak{M}}(x_1, \dots, x_m) : \varphi \text{ is a formula of } \mathfrak{L}_\epsilon \}$ is already closed under functional composition by definability, and hence is a complete set of Skolem functions for $\mathfrak{M}$.
So here are my questions:
Q1: Should'nt we consider the Skolem functions for each different context of each formula $\varphi$?Because I think if we fix only one context, then the other free variable won't be accounted for when placing existential quantifiers in arbitrary orders behind them.
Q2: I don't understand how definablity plays a role in showing these functions are closed under composition. For example what would the desired Skolem function for $t_{\varphi}^{\mathfrak{M}}(t_{\psi}^{\mathfrak{M}}(y_1, \dots, y_n), x_1, \dots, x_m)$ look like?