Example 58.3.5. Let \mathcal{C} be a category and let F : \mathcal{C} \to \textit{Sets} be a functor such that F(X) is finite for all X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}). By Lemma 58.3.1 we see that G = \text{Aut}(F) comes endowed with the structure of a profinite topological group in a canonical manner. We obtain a functor
where F(X) is endowed with the induced action of G. This action is continuous by our construction of the topology on \text{Aut}(F).
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