The Stacks project

Proof. First proof: Combine Lemmas 20.49.5, 20.47.4, and 20.48.6. Second proof (sketch): Say $K$ and $L$ are perfect. After replacing $X$ by the members of an open covering we may assume that $K$ and $L$ are represented by strictly perfect complexes $\mathcal{K}^\bullet $ and $\mathcal{L}^\bullet $. After replacing $X$ by the members of an open covering we may assume the map $K \to L$ is given by a map of complexes $\alpha : \mathcal{K}^\bullet \to \mathcal{L}^\bullet $, see Lemma 20.46.8. Then $M$ is isomorphic to the cone of $\alpha $ which is strictly perfect by Lemma 20.46.2. $\square$