The Stacks project

Proof. Combinging Lemma 77.13.3 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 it follows from More on Morphisms of Spaces, Lemma 76.51.4 that $K$ is pseudo-coherent relative to $A$. Since $X$ is of flat and of finite presentation over $A$, this is the same as being pseudo-coherent on $X$, see More on Morphisms of Spaces, Lemma 76.45.4. $\square$