The Stacks project

Definition 4.42.3. 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

is representable.