The Stacks project

Lemma 8.8.2. Let $\mathcal{C}$ be a site. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category over $\mathcal{C}$. Let $p' : \mathcal{S}' \to \mathcal{C}$ and $G : \mathcal{S} \to \mathcal{S}'$ the stack and $1$-morphism constructed in Lemma 8.8.1. This construction has the following universal property: Given a stack $q : \mathcal{X} \to \mathcal{C}$ and a $1$-morphism $F : \mathcal{S} \to \mathcal{X}$ of fibred 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. Omitted. Hint: Suppose that $x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}'_ U)$. By the result of Lemma 8.8.1 there exists a covering $\{ U_ i \to U\} _{i \in I}$ such that $x'|_{U_ i} = G(x_ i)$ for some $x_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{U_ i})$. Moreover, there exist coverings $\{ U_{ijk} \to U_ i \times _ U U_ j\} $ and isomorphisms $\alpha _{ijk} : x_ i|_{U_{ijk}} \to x_ j|_{U_{ijk}}$ with $G(\alpha _{ijk}) = \text{id}_{x'|_{U_{ijk}}}$. Set $y_ i = F(x_ i)$. Then you can check that

\[ F(\alpha _{ijk}) : y_ i|_{U_{ijk}} \to y_ j|_{U_{ijk}} \]

agree on overlaps and therefore (as $\mathcal{X}$ is a stack) define a morphism $\beta _{ij} : y_ i|_{U_ i \times _ U U_ j} \to y_ j|_{U_ i \times _ U U_ j}$. Next, you check that the $\beta _{ij}$ define a descent datum. Since $\mathcal{X}$ is a stack these descent data are effective and we find an object $y$ of $\mathcal{X}_ U$ agreeing with $G(x_ i)$ over $U_ i$. The hint is to set $H(x') = y$. $\square$


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