The Stacks project

Lemma 18.21.5. With $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, $s : \mathcal{G} \to \mathcal{F}$ as in Lemma 18.21.4. If there exist a morphism $f : V \to U$ of $\mathcal{C}$ such that $\mathcal{G} = h_ V^\# $ and $\mathcal{F} = h_ U^\# $ and $s$ is induced by $f$, then the diagrams of Lemma 18.19.5 and Lemma 18.21.4 agree via the identifications $(j_\mathcal {F}, j_\mathcal {F}^\sharp ) = (j_ U, j_ U^\sharp )$ and $(j_\mathcal {G}, j_\mathcal {G}^\sharp ) = (j_ V, j_ V^\sharp )$ of Lemma 18.21.3.

Proof. All assertions follow from Sites, Lemma 7.30.7 except for the assertion that the two maps $j^\sharp $ agree. This holds since in both cases the map $j^\sharp $ is simply the identity. Some details omitted. $\square$

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