The Stacks project

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

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

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

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

  2. 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.

Proof. The first assertion follows immediately from Lemma 4.39.5. For the second, suppose that $j' : \mathcal{S} \to \mathcal{C}/X'$ is a second such pair. Choose a $1$-morphism $t' : \mathcal{C}/X' \to \mathcal{S}$ such that $j' \circ t' \cong \text{id}_{\mathcal{C}/X'}$ and $t' \circ j' \cong \text{id}_\mathcal {S}$. Then $j \circ t' : \mathcal{C}/X' \to \mathcal{C}/X$ is an equivalence. Hence it is an isomorphism, see Lemma 4.38.6. Hence by the Yoneda Lemma 4.3.5 (via Example 4.38.7 for example) it is given by an isomorphism $X' \to X$. $\square$

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