Lemma 78.12.3 (0442)—The Stacks project

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Part 4: Algebraic Spaces
Chapter 78: Groupoids in Algebraic Spaces
Section 78.12: Quasi-coherent sheaves on groupoids
Lemma 78.12.3 (cite)

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Lemma 78.12.3. Let $B \to S$ as in Section 78.3. Consider a morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoid in algebraic spaces over $B$. Then pullback $f^*$ given by

\[ (\mathcal{F}, \alpha ) \mapsto (f^*\mathcal{F}, f^*\alpha ) \]

defines a functor from the category of quasi-coherent sheaves on $(U', R', s', t', c')$ to the category of quasi-coherent sheaves on $(U, R, s, t, c)$.

Proof. Omitted. $\square$

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