Lemma 7.26.3. Let $\mathcal{C}$ be a site and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (h_ U^\# , \mathcal{F}) = j_*(\mathcal{F}|_{\mathcal{C}/U})$ for $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Proof. This can be shown by directly constructing an isomorphism of sheaves. Instead we argue as follows. Let $\mathcal{G}$ be a sheaf on $\mathcal{C}$. Then

\begin{align*} \mathop{Mor}\nolimits (\mathcal{G}, j_*(\mathcal{F}|_{\mathcal{C}/U})) & = \mathop{Mor}\nolimits (\mathcal{G}|_{\mathcal{C}/U}, \mathcal{F}|_{\mathcal{C}/U}) \\ & = \mathop{Mor}\nolimits (j_!(\mathcal{G}|_{\mathcal{C}/U}), \mathcal{F}) \\ & = \mathop{Mor}\nolimits (\mathcal{G} \times h_ U^\# , \mathcal{F}) \\ & = \mathop{Mor}\nolimits (\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (h_ U^\# , \mathcal{F})) \end{align*}

and we conclude by the Yoneda lemma. Here we used Lemmas 7.26.2 and 7.25.7. $\square$

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