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"](https://eudml.org/doc/89765). 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.



Objects and morphisms in inverse limits of toposes?
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](http://library.msri.org/books/sga/sga/pdf/sga4-1.pdf)). 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"](https://repository.ubn.ru.nl/handle/2066/129084).


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 $\pi\_{jk}(H'')=H'$.


