The Stacks project

Remark 18.27.10. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf of sets on $\mathcal{C}$ and consider the localization morphism $j : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. See Sites, Definition 7.30.4. We claim that (a) $j_!\mathbf{Z} = \mathbf{Z}_\mathcal {F}^\# $ and (b) $j_!(j^{-1}\mathcal{H}) = j_!\mathbf{Z} \otimes _\mathbf {Z} \mathcal{H}$ for any abelian sheaf $\mathcal{H}$ on $\mathcal{C}$. Let $\mathcal{G}$ be an abelian on $\mathcal{C}$. Part (a) follows from the Yoneda lemma because

\[ \mathop{\mathrm{Hom}}\nolimits (j_!\mathbf{Z}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, j^{-1}\mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}_\mathcal {F}^\# , \mathcal{G}) \]

where the second equality holds because both sides of the equality evaluate to the set of maps from $\mathcal{F} \to \mathcal{G}$ viewed as an abelian group. For (b) we use the Yoneda lemma and

\begin{align*} \mathop{\mathrm{Hom}}\nolimits (j_!(j^{-1}\mathcal{H}), \mathcal{G}) & = \mathop{\mathrm{Hom}}\nolimits (j^{-1}\mathcal{H}, j^{-1}\mathcal{G}) \\ & = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (j^{-1}\mathcal{H}, j^{-1}\mathcal{G})) \\ & = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, j^{-1}\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{H}, \mathcal{G})) \\ & = \mathop{\mathrm{Hom}}\nolimits (j_!\mathbf{Z}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{H}, \mathcal{G})) \\ & = \mathop{\mathrm{Hom}}\nolimits (j_!\mathbf{Z} \otimes _\mathbf {Z} \mathcal{H}, \mathcal{G}) \end{align*}

Here we use adjunction, the fact that taking $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ commutes with localization, and Lemma 18.27.6.

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