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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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