Lemma 7.13.4. In the situation of Lemma 7.13.2. For any presheaf $\mathcal{G}$ on $\mathcal{C}$ we have $(u_ p\mathcal{G})^\# = (u_ p(\mathcal{G}^\# ))^\#$.

Proof. For any sheaf $\mathcal{F}$ on $\mathcal{D}$ we have

\begin{eqnarray*} \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(u_ s(\mathcal{G}^\# ), \mathcal{F}) & = & \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}^\# , u^ s\mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{G}^\# , u^ p\mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{G}, u^ p\mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{D})}(u_ p\mathcal{G}, \mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}((u_ p\mathcal{G})^\# , \mathcal{F}) \end{eqnarray*}

and the result follows from the Yoneda lemma. $\square$

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