Definition 4.41.5. 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. We say $F$ is representable, or that $\mathcal{X}$ is relatively representable over $\mathcal{Y}$, if for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any $G : \mathcal{C}/U \to \mathcal{Y}$ the category fibred in groupoids

$(\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X} \longrightarrow \mathcal{C}/U$

is representable.

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