The Stacks project

Remark 4.41.3. We can use the $2$-Yoneda lemma to give an alternative proof of Lemma 4.37.3. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. We define a contravariant functor $F$ from $\mathcal{C}$ to the category of groupoids as follows: for $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ let

\[ F(U) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}). \]

If $f : U \to V$ the induced functor $\mathcal{C}/U \to \mathcal{C}/V$ induces the morphism $F(f) : F(V) \to F(U)$. Clearly $F$ is a functor. Let $\mathcal{S}'$ be the associated category fibred in groupoids from Example 4.37.1. There is an obvious functor $G : \mathcal{S}' \to \mathcal{S}$ over $\mathcal{C}$ given by taking the pair $(U, x)$, where $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x \in F(U)$, to $x(\text{id}_ U) \in \mathcal{S}$. Now Lemma 4.41.2 implies that for each $U$,

\[ G_ U : \mathcal{S}'_ U = F(U)= \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}) \to \mathcal{S}_ U \]

is an equivalence, and thus $G$ is an equivalence between $\mathcal{S}$ and $\mathcal{S}'$ by Lemma 4.35.9.

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