Lemma 76.52.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $g : Y' \to Y$ be a morphism of algebraic spaces over $S$. Set $X' = Y' \times _ Y X$ and denote $g' : X' \to X$ the projection. If $K \in D(\mathcal{O}_ X)$ is $Y$-perfect, then $L(g')^*K$ is $Y'$-perfect.

**Proof.**
Reduce to the case of schemes using Lemma 76.52.3 and then apply Derived Categories of Schemes, Lemma 36.35.6.
$\square$

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