The Stacks project

Lemma 7.30.7. Assume $\mathcal{C}$ and $s : \mathcal{G} \to \mathcal{F}$ are as in Lemma 7.30.6. If $\mathcal{G} = h_ V^\# $ and $\mathcal{F} = h_ U^\# $ and $s : \mathcal{G} \to \mathcal{F}$ comes from a morphism $V \to U$ of $\mathcal{C}$ then the diagram in Lemma 7.30.6 is identified with diagram (7.25.8.1) via the identifications $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ V^\# $ and $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ of Lemma 7.25.4.

Proof. This is true because the descriptions of $j^{-1}$ agree. See Lemma 7.25.9 and Lemma 7.30.6. $\square$

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