The Stacks project

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
\begin{equation} \label{pione-equation-remember} \mathcal{C} \longrightarrow \textit{Finite-}G\textit{-Sets},\quad X \longmapsto F(X) \end{equation}

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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