I have a simple question on notation.

Let $S$ be a Henselian trait with closed point $s$ (with finite residue field $k$) and generic point $\eta$. Let $X/S$ be a variety. Then, we have the functor

$$R\Psi:D^b_c((X_\eta)_\mathrm{\acute{e}t},\overline{\mathbb{Q}_\ell})\to D^b_c(X_s\times_s \eta,\overline{\mathbb{Q}_\ell})$$

where $D^b_c(X_s\times_s\eta,\overline{\mathbb{Q}_\ell})$ denotes the category of constructible $\overline{\mathbb{Q}_\ell}$-sheaves on $X_{\overline{s}}$ with "an action of $\mathrm{Gal}(\overline{\eta}/\eta)$ compatible with the action of $\mathrm{Gal}(\overline{s}/s)$." This means (a la "Le Formalisme de Cycles Evanescents" in SGA 7) if, for example, we're dealing with a constructible $\overline{\mathbb{Q}_\ell}$-sheaf $\mathcal{F}$ on $X_{\overline{s}}$, that for all $g\in \mathrm{Gal}(\overline{\eta}/\eta)$ we have isomorphisms:

$$\sigma(g):\overline{g}_\ast \mathcal{F}\to\mathcal{F}$$ (where $\overline{g}\in\mathrm{Gal}(\overline{s}/s)$) such that $\sigma(gh)=\sigma(g)\sigma(h)$.

Something which I commonly see is the following. People say that for $x\in X_s(k)$ that considering "$(R\Psi\overline{\mathbb{Q}_\ell})_x$" one gets an element of the derived category of finite-dimensional $\mathrm{Gal}(\overline{\eta}/\eta)$-representations.

**Questions:**

1) What does $(R\Psi\overline{\mathbb{Q}_\ell})_x$ even mean? This doesn't make any literal sense to me. Here are two possibilities I've considered:

a) If I forget the extra structure of the $\mathrm{Gal}(\overline{\eta}/\eta)$ action, it doesn't make sense (unless I am being silly) to take the stalk an $\overline{\mathbb{Q}_\ell}$-sheaf at a point $x\in X_s(k)$. One could interpret it at the choice of $\overline{\eta}$ gives you a canonical $\overline{x}\in X_{\overline{s}}(\overline{k})$ and so $(R\Psi\overline{\mathbb{Q}_\ell})_x$ might be shorthand for $(R\Psi\overline{\mathbb{Q}_\ell})_{\overline{x}}$. If that's the case, I don't see why it's $\mathrm{Gal}(\overline{\eta}/\eta)$-stable, so that one actually gets an action of $\mathrm{Gal}(\overline{\eta}/\eta)$.

b) Similar to a), but instead of taking the stalk $R\Psi\overline{\mathbb{Q}_\ell}$ thought about as an element of $D^b_c((X_s)_\mathrm{\acute{e}t},\overline{\mathbb{Q}_\ell})$ take the canonical pair $(\overline{x},\overline{\eta})$ (a point of the topos $X_s\times_s \eta$) and consider the stalk of this point. This gives the cohomology of the 'Milnor fiber', which also seems wrong. (**EDIT:** Ignore b) People think I'm claiming that the stalk at the point $(\overline{x},\overline{\eta})$ is not the cohomology of the Milnor fiber--I know this to be true. What I meant to say, even though I highly doubted it, that the stalk at $(\overline{x},\overline{\eta})$ was another interpretation of $(R\Psi\overline{\mathbb{Q}_\ell})_x$ but I very much don't think that now.)

2) Once I figure out what $(R\Psi\overline{\mathbb{Q}_\ell})_x$ means, how is it a finite dimensional continuous $\mathrm{Gal}(\overline{\eta}/\eta)$-representation (if not obvious from the definition).

Thanks so much!

**EDIT:** As examples of this notation see the second to last paragraph on page 13 of this article or Theorem 7.10 of this article.