The Stacks project

Lemma 8.9.2. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids over $\mathcal{C}$. Let $p' : \mathcal{S}' \to \mathcal{C}$ and $G : \mathcal{S} \to \mathcal{S}'$ the stack in groupoids and $1$-morphism constructed in Lemma 8.9.1. This construction has the following universal property: Given a stack in groupoids $q : \mathcal{X} \to \mathcal{C}$ and a $1$-morphism $F : \mathcal{S} \to \mathcal{X}$ of categories over $\mathcal{C}$ there exists a $1$-morphism $H : \mathcal{S}' \to \mathcal{X}$ such that the diagram

\[ \xymatrix{ \mathcal{S} \ar[rr]_ F \ar[rd]_ G & & \mathcal{X} \\ & \mathcal{S}' \ar[ru]_ H } \]

is $2$-commutative.

Proof. This is a special case of Lemma 8.8.2. $\square$

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