If $f$ is flat, $\mathcal{F}, \mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\mathit{QCoh}(\mathcal{O}_\mathcal {X}))$, and $\mathcal{F}$ of finite presentation and let then we have
\[ d(hom(\mathcal{F}, \mathcal{G})) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(d(\mathcal{F}), d(\mathcal{G})) \]with notation as in Lemma 103.10.8. Perhaps the easiest way to see this is as follows
\begin{align*} d(hom(\mathcal{F}, \mathcal{G})) & = d(Q(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G}))) \\ & = c(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})) \\ & = f^*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})|_{U_{\acute{e}tale}} \\ & = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {U}}(f^*\mathcal{F}, f^*\mathcal{G})|_{U_{\acute{e}tale}} \\ & = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(f^*\mathcal{F}|_{U_{\acute{e}tale}}, f^*\mathcal{G}|_{U_{\acute{e}tale}}) \end{align*}The first equality by construction of $hom$. The second equality by (7). The third equality by definition of $c$. The fourth equality by Modules on Sites, Lemma 18.31.4. The final equality by the same reference applied to the flat morphism of ringed topoi $i_ U (U_{\acute{e}tale}, \mathcal{O}_ U) \to (\mathcal{U}_{\acute{e}tale}, \mathcal{O}_\mathcal {U})$ of Sheaves on Stacks, Lemma 96.10.1.
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