The Stacks project

Lemma 4.41.7. Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. Make a choice of pullbacks for $\mathcal{Y}$. Assume

  1. each functor $F_ U : \mathcal{X}_ U \longrightarrow \mathcal{Y}_ U$ between fibre categories is faithful, and

  2. for each $U$ and each $y \in \mathcal{Y}_ U$ the presheaf

    \[ (f : V \to U) \longmapsto \{ (x, \phi ) \mid x \in \mathcal{X}_ V, \phi : f^*y \to F(x)\} /\cong \]

    is a representable presheaf on $\mathcal{C}/U$.

Then $F$ is representable.

Proof. Clear from the description of fibre categories in Lemma 4.41.3 and the characterization of representable fibred categories in Lemma 4.40.2. $\square$


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