Let $F : \mathcal{C} \to \textit{Groupoids}$ be a functor. There is a category cofibered in groupoids $\mathcal{F} \to \mathcal{C}$ associated to $F$ as follows. An object of $\mathcal{F}$ is a pair $(U, x)$ where $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x \in \mathop{\mathrm{Ob}}\nolimits (F(U))$. A morphism $(U, x) \to (V, y)$ is a pair $(f, a)$ where $f \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, V)$ and $a \in \mathop{\mathrm{Mor}}\nolimits _{F(V)}(F(f)(x), y)$. The functor $\mathcal{F} \to \mathcal{C}$ sends $(U, x)$ to $U$. See Categories, Section 4.37.

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