The Stacks project

Lemma 18.30.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be a quasi-compact object of $\mathcal{C}$. Then the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, -)$ commutes with direct sums.

Proof. This is true because $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, \mathcal{F}) = \mathcal{F}(U)$ by (18.19.2.1) and because the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ commutes with direct sums by Lemma 18.30.3. $\square$

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