Lemma 4.40.2. Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids.

$\mathcal{S}$ is representable if and only if the following conditions are satisfied:

$\mathcal{S}$ is fibred in setoids, and

the presheaf $U \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)/\cong $ is representable.

If $\mathcal{S}$ is representable the pair $(X, j)$, where $j$ is the equivalence $j : \mathcal{S} \to \mathcal{C}/X$, is uniquely determined up to isomorphism.

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