Objects and morphisms in inverse limits of toposes? Certain Galois toposes can be written as $\lim_{i \in I} \mathbf{PSh}(G_i)$ where $(G_i)_{i \in I}$ is an inverse system of discrete groups. (The limit is a strict limit in the 2-category of Grothendieck toposes and geometric morphisms.)
What are the objects and morphisms in this topos?
If all groups $G_i$ in the diagram are finite and the transition maps are surjective, then I believe the objects of the topos are sets $S$ together with a $G$-action, with the special property that the action morphism $G \to \mathrm{End}(S)$ factors through one of the groups $G_i$. The morphisms are the functions such that $\phi(g\cdot x) = g\cdot \phi(x)$ for all $g\in G$. Can you do something similar in general?
Edit: One of the standard references for this is "Continuous fibrations and inverse limits of toposes". The idea there is to replace the diagram of Grothendieck toposes by a corresponding diagram of Grothendieck sites, with a special kind of transition maps (called continuous fibrations). Then a "limit Grothendieck site" is constructed and the inverse limit topos is the topos of sheaves on this site. But I don't know how to follow this process in practice, even in the case described above where all groups are finite and the transition maps between the groups are surjective.
 A: The question was solved thanks to comments by Marc Hoyois.
The classifying topos associated to a pro-group is described by Grothendieck and Verdier in SGA4, Exposé IV, 2.7 (link). Grothendieck and Verdier assume that the transition maps $\pi_{ij} : G_j \to G_i$ are surjective.
The classifying topos of a pro-group $(G_i)_{i \in I}$ as defined there, agrees with the inverse limit topos $\varprojlim_{i \in I} \mathbf{PSh}(G_i)$, by Remark 2.7 in "Prodiscrete groups and Galois toposes".
As suggested by მამუკა ჯიბლაძე below, here the construction of Grothendieck and Verdier in more detail:
The objects of $\varprojlim_{i \in I} \mathbf{PSh}(G_i)$ are sets $E$, written as directed union $\bigcup_{i \in I} E_i$ where each $E_i$ comes with a $G_i$-action, and such that $E_i = \{ x \in E_j : gx = x,~\forall g \in \ker(G_j \to G_i)\}$ and the action of $G_i$ on $E_i$ is the one induced by the action of $G_j$ on $E_j$ (this last part is left implicit in SGA4).
If we write $E_j$ as a direct union of orbits, then $E_i$ consists of the orbits isomorphic to $G_j/H$ with $H$ containing $\ker(G_j \to G_i)$. In this way, we get another description of the objects of $\varprojlim_{i \in I} \mathbf{PSh}(G_i)$. They are isomorphic to direct sums of quotients $G_i/H$, where two quotients $G_i/H$ and $G_j/H'$ are the same if there is a $k \leq i,j$ and a subgroup $H'' \subseteq G_k$ such that $\pi_{ik}(H'') = H$ and $\pi_{jk}(H'')=H'$.
