Lemma 18.27.6. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings.

1. Let $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ be presheaves of $\mathcal{O}$-modules. There is a canonical isomorphism

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}, \mathcal{H}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{H}))$

which is functorial in all three entries (sheaf Hom in all three spots). In particular,

$\mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}, \mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{H}))$
2. Suppose that $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, and $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ are sheaves of $\mathcal{O}$-modules. There is a canonical isomorphism

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{F} \otimes _\mathcal {O} \mathcal{G}, \mathcal{H}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{H}))$

which is functorial in all three entries (sheaf Hom in all three spots). In particular,

$\mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}( \mathcal{F} \otimes _\mathcal {O} \mathcal{G}, \mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}( \mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{H}))$

Proof. This is the analogue of Algebra, Lemma 10.12.8. The proof is the same, and is omitted. $\square$

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