Definition 8.5.5. Let $\mathcal{C}$ be a site. The *$2$-category of stacks in groupoids over $\mathcal{C}$* is the sub $2$-category of the $2$-category of stacks over $\mathcal{C}$ (see Definition 8.4.5) defined as follows:

Its objects will be stacks in groupoids $p : \mathcal{S} \to \mathcal{C}$.

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 every functor preserves them.)

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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