Theorem 38.29.6 (Derived Grothendieck Existence Theorem). In Situation 38.29.1 there exists a pseudo-coherent $K$ in $D(\mathcal{O}_ X)$ such that $K_ n = K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_{X_ n}$ for all $n$ compatibly with the maps $\varphi _ n$.

**Proof.**
Apply Lemmas 38.29.2, 38.29.3, 38.29.4 to get a pseudo-coherent object $K$ of $D(\mathcal{O}_ X)$. Choosing affine opens in Lemma 38.29.5 it follows immediately that $K$ restricts to $K_ n$ over $X_ n$.
$\square$

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