I'm currently reading some notes on AECs (here), and just wanted to make sure I understand the argument in Corollary 12.8, for I'm still getting used to generalized Ehrenfeucht-Mostowski models in AECs. As far as I understand, EM blueprints/templates $\Phi$ keep track of the quantifier-free sentences some indiscernible satisfies (that is, a single $n$-type for each $n < \omega$) and are defined as such assuming we can find indiscernibles satisfying them inside some $\tau(\Phi)$-structure. Finding them makes use of the presentation theorem and Ramsey's Theorem/Erdos-Rado. Given one $\Phi$ we one can build models $\operatorname{EM}(I, \Phi)$ by taking the $\tau(\Phi)$-hull of the indiscernibles inside.
Moving on to the mentioned argument, from the definition each $\Phi$ has an associated vocabulary, and the author states that given some blueprint $\Psi$ we can expand the vocabulary to get $\Phi$ s.t. $\operatorname{EM}_\tau(I, \Phi) = \operatorname{EM}_\tau(I^{< \omega}, \Psi)$. Now, since there are no further details, I was wondering if adding $n$-ary $\sigma_n(i_1, \dots, i_n)$ to the vocabulary to represent $I^{< \omega}$ along with some quantifier-free sentences in $\Psi$ witnessing the indiscernibility of the obtained $I^{< \omega}$ from the $\sigma_n$ would be enough/work. how one would define such $\Phi$.
I'd appreciate if someone familiarized with AECs could check for me and in case that's nonsense, give some details for this template construction argument. I'd also appreciate if someone could point out any mistakes in my understanding of the definition and use of EM templates/blueprints.
Any help is appreciated!
EDIT: "Adding quantifier-free sentences in $\Psi$ witnessing the indiscernibility of the obtained $I^{< \omega}$" seems to be quite naive now that I think about it. Any two increasing sequences in $I^{< \omega}$ need not have same order type upon being expanded (that is, concatenating the elements of the sequences), so no guarantee we'd be obtaining indiscernibles by this method. Similarly, given any sequence, if I wanted to organize it as a sequence in $I^{<\omega}$ by grouping the elements, I could obtain many different order types.