Lemma 77.12.4. In Situation 77.12.1 let $K$ be as in Lemma 77.12.2. Then $K$ is pseudo-coherent relative to $A$.

**Proof.**
Combinging Lemma 77.12.3 and Derived Categories of Spaces, Lemma 75.25.7 we see that $R\Gamma (X, K \otimes ^\mathbf {L} E)$ is pseudo-coherent in $D(A)$ for all pseudo-coherent $E$ in $D(\mathcal{O}_ X)$. Thus the lemma follows from More on Morphisms of Spaces, Lemma 76.51.4.
$\square$

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