The Stacks project

Remark 62.4.9. Let $g : Y' \to Y$ be a morphism of schemes. For an abelian presheaf $\mathcal{G}'$ on $Y'_{\acute{e}tale}$ let us denote $g_*\mathcal{G}'$ the presheaf $V \mapsto \mathcal{G}'(Y' \times _ Y V)$. If $\alpha : \mathcal{G} \to g_*\mathcal{G}'$ is a map of abelian presheaves on $Y_{\acute{e}tale}$, then there is a unique map $\alpha ^\# : \mathcal{G}^\# \to g_*((\mathcal{G}')^\# )$ of abelian sheaves on $Y_{\acute{e}tale}$ such that the diagram

\[ \xymatrix{ \mathcal{G} \ar[d] \ar[r]_\alpha & g_*\mathcal{G}' \ar[d] \\ \mathcal{G}^\# \ar[r]^-{\alpha ^\# } & g_*((\mathcal{G}')^\# ) } \]

is commutative where the vertical maps come from the canonical maps $\mathcal{G} \to \mathcal{G}^\# $ and $\mathcal{G}' \to (\mathcal{G}')^\# $. If $\alpha ' : g^{-1}\mathcal{G}^\# \to (\mathcal{G}')^\# $ is the map adjoint to $\alpha ^\# $, then for a geometric point $\overline{y}' : \mathop{\mathrm{Spec}}(k) \to Y'$ with image $\overline{y} = g \circ \overline{y}'$ in $Y$, the map

\[ \alpha '_{\overline{y}'} : \mathcal{G}_{\overline{y}} = (\mathcal{G}^\# )_{\overline{y}} = (g^{-1}\mathcal{G}^\# )_{\overline{y}'} \longrightarrow (\mathcal{G}')^\# _{\overline{y}'} = \mathcal{G}'_{\overline{y}'} \]

is given by mapping the class in the stalk of a section $s$ of $\mathcal{G}$ over an étale neighbourhood $(V, \overline{v})$ to the class of the section $\alpha (s)$ in $g_*\mathcal{G}'(V) = \mathcal{G}'(Y' \times _ Y V)$ over the étale neighbourhood $(Y' \times _ Y V, (\overline{y}', \overline{v}))$ in the stalk of $\mathcal{G}'$ at $\overline{y}'$.


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