My question is inspired by definitions 3.1 and 3.2 (3) of "Double helix in large large cardinals and iteration of elementary embeddings" by Kentaro Sato 2007 but I think my question is relevant to what most set theorists call extenders, despite Sato's unusual terminology. The relevant part of definition 3.1 is:
For ordinals $\mu \le \lambda$, partition space(s) [...] $\mathcal{E}_{\lambda,\mu}$ [is] defined as follows: $$\mathit{P} \in \mathcal{E}_{\lambda,\mu} \iff (\exists \mathit{a} \subset \lambda)(\vert\mathit{a}\vert \lt \omega \And \mathit{P} \sqsupset \{\{\mathit{s} \in D(\mathcal{E}_{\lambda,\mu} \mid \mathit{s}\upharpoonright\mathit{a} = \mathit{f} \} \mid \mathit{f} \in ^\mathit{a}\mu \})$$
Why are the sets $\mathit{a}$ (which determine whether functions $\mathit{s}$ belong to a given $\mathit{E}$-large set for some ultrafilter $\mathit{E}$) restricted to be finite? With higher bounds on the size of $\mathit{a}$ I can probably prove closure properties of ultrapowers by ultrafilters of $\mathcal{E}_{\lambda,\mu}$ (extenders), which would be useful for placing ultrahuge cardinals in the large cardinal hierarchy, but with the bound $\omega$ it's not obvious to me that the ultrapower is even countably closed.