The Stacks project

Lemma 18.27.8. Let $\mathcal{C}$ be a category, resp. a site Let $\mathcal{O} \to \mathcal{O}'$ be a map of presheaves, resp. sheaves of rings. Then

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}', \mathcal{F})) \]

for any $\mathcal{O}'$-module $\mathcal{G}$ and $\mathcal{O}$-module $\mathcal{F}$.

Proof. This is the analogue of Algebra, Lemma 10.14.4. The proof is the same, and is omitted. $\square$

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