Does base change respect Galois correspondence between $\ell$-adic sheaves and representations of the fundamental étale group?

Question

It is known that for $X$ a connected scheme there is an equivalence of categories

$$\left\lbrace \text{$\ell$-adic smooth sheaves over $X$} \right\rbrace \leftrightarrow \left\lbrace \text{$\ell$-adic representations of $\pi_{1,et}(X,\overline{x})$} \right\rbrace,$$

but is there any chance that there is a commutative square for the base change, that is (up to hypothesis on $S$) a commutative diagram

$$\begin{array}{ccc} \left\lbrace \text{$\ell$-adic smooth sheaves over $X$} \right\rbrace & \leftrightarrow & \left\lbrace \text{$\ell$-adic representations of $\pi_{1,et}(X,\overline{x})$} \right\rbrace \\ \downarrow & & \downarrow \\ \left\lbrace \text{$\ell$-adic smooth sheaves over $X\times S$} \right\rbrace & \leftrightarrow & \left\lbrace \text{$\ell$-adic representations of $\pi_{1,et}(X\times S,\overline{x'})$} \right\rbrace, \end{array}$$

The existence of vertical arrows is obvious, but since the theory of Galois category is very abstract I have no idea if the Galois correspondence is respected by base change.

Answer 1

Your rightmost arrow comes from the map on fundamental groups. Where does that map even come from in the étale theory? It comes from the pullback functor between categories of finite étale covers and https://stacks.math.columbia.edu/tag/0BN5 which gives a criterion for a functor between Galois categories to induce such a map. But the defining property of the map is a commutative diagram which looks a lot like the one you want.

What is needed to transform the defining commutative diagram you have into the one you want?

One first needs to check that the functor from finite étale covers to sheaves via their sheaves of sections is compatible with the pullback functors of the two categories. This is an exercise with the definition of pullback sheaf.

One also has to follow the steps of the construction of $\ell$-adic sheaves from sheaves of finite sets - i.e. to check commutativity of the diagram on sheaves of finite sets and $\pi_1$-sets implies commutativity on sheaves of finite groups and $\pi_1$-representations since these are just the group objects in the relevant categories, and then check that commutativity for mod $\ell^n$ sheaves implies commutativity for $\ell$-adic sheaves by a limiting argument.

Note that this has nothing to do with base change and just applies to pullback on an arbitrary map.