The Stacks project

Lemma 99.11.2. In Situation 99.11.1 the functor $\mathrm{Pic}_{X/B}$ is the sheafification of the functor $T \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{P}\! \mathit{ic}_{X/B, T})/\cong $.

Proof. Since the fibre category $\mathcal{P}\! \mathit{ic}_{X/B, T}$ of the Picard stack $\mathcal{P}\! \mathit{ic}_{X/B}$ over $T$ is the category of invertible sheaves on $X_ T$ (see Section 99.10 and Examples of Stacks, Section 95.16) this is immediate from the definitions. $\square$

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