The Stacks project

Definition 4.38.3. Let $\mathcal{C}$ be a category. The $2$-category of categories fibred in sets over $\mathcal{C}$ is the sub $2$-category of the category of categories fibred in groupoids over $\mathcal{C}$ (see Definition 4.35.6) defined as follows:

  1. Its objects will be categories $p : \mathcal{S} \to \mathcal{C}$ fibred in sets.

  2. Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$ (since every morphism is strongly cartesian $G$ automatically preserves them).

  3. Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_ x) = \text{id}_{p(x)}$ for all $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S})$.

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