Lemma 109.60.2. Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{F}_ n$ is a collection of locally quasi-coherent sheaves with the flat base change property on $\mathcal{X}$, then $\oplus _ n \mathcal{F}_ n[n] \to \prod _ n \mathcal{F}_ n[n]$ is an isomorphism in $D(\mathcal{O}_\mathcal {X})$.

**Proof.**
This is true because by Lemma 109.60.1 we see that the direct sum is isomorphic to the product.
$\square$

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