The Stacks project

Proof. Let $U$ be an object of $\mathcal{C}$. Since the topology on $\mathcal{C}$ is chaotic, the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ is exact and commutes with direct sums. Hence the exact functor $M \mapsto R\Gamma (U, M)$ is computed by representing $K$ by any differential graded $\mathcal{A}$-module $\mathcal{M}$ and taking $\mathcal{M}(U)$. Thus $R\Gamma (U, -)$ commutes with direct sums, see Lemma 24.26.8. Similarly, given a morphism $U \to V$ of $\mathcal{C}$ the derived tensor product functor $- \otimes _{\mathcal{O}(A)}^\mathbf {L} \mathcal{A}(U) : D(\mathcal{A}(V)) \to D(\mathcal{A}(U))$ is exact and commutes with direct sums. The lemma follows from these observations in a straightforward manner; details omitted. $\square$