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.

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