Lemma 8.5.4 (042X)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 8: Stacks
Section 8.5: Stacks in groupoids
Lemma 8.5.4 (cite)

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Lemma 8.5.4. Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$ , $\mathcal{S}_2$ be categories over $\mathcal{C}$ . Suppose that $\mathcal{S}_1$ and $\mathcal{S}_2$ are equivalent as categories over $\mathcal{C}$ . Then $\mathcal{S}_1$ is a stack in groupoids over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is a stack in groupoids over $\mathcal{C}$ .

Proof. Follows by combining Lemmas 8.5.2 and 8.4.4. $\square$

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