The Stacks project

Example 4.35.4. A homomorphism of groups $p : G \to H$ gives rise to a functor $p : \mathcal{S}\to \mathcal{C}$ as in Example 4.2.12. This functor $p : \mathcal{S}\to \mathcal{C}$ is fibred in groupoids if and only if $p$ is surjective. The fibre category $\mathcal{S}_ U$ over the (unique) object $U\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is the category associated to the kernel of $p$ as in Example 4.2.6.