Lemma 11.2.2 (cf. Hovey et al. (1997)). Let \(C\) be a presentably symmetric monoidal stable \(\infty \)-category in the sense of Definition 22.5.1, and let \(\Gg \) be a small collection of compact generators of \(C\). If every object of \(\Gg \) is dualizable, then every compact object of \(C\) is dualizable.
Proof. The tensor product admits internal homs by Theorem 22.2.5. The result on compact generation, Proposition 22.3.4 from Chapter 22, which we use as a black box, identifies the compact objects of \(C\) with the thick subcategory generated by \(\Gg \). The claim now follows from Lemma 11.1.16. □
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