Lemma 8.13.2. Assume that $\mathcal{C}$ is a site, and $U$ is an object of $\mathcal{C}$ whose associated representable presheaf is a sheaf. Constructions A and B above define mutually inverse (!) functors of $2$-categories

\[ \left\{ \begin{matrix} 2\text{-category of}
\\ \text{stacks over }\mathcal{C}/U
\end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} 2\text{-category of pairs }(\mathcal{T}, p) \text{ consisting}
\\ \text{of a stack }\mathcal{T}\text{ over }\mathcal{C}\text{ and a morphism}
\\ p : \mathcal{T} \to \mathcal{C}/U\text{ of stacks over }\mathcal{C}
\end{matrix} \right\} \]

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