Lemma 11.1.16. Let \(C\) be a stably symmetric monoidal \(\infty \)-category which admits internal homs. Then the collection of dualizable objects forms a thick subcategory of \(C\), i.e. it is a stable subcategory closed under retracts.
Proof. By Corollary 11.1.13, an object \(X\) is dualizable if and only if the natural map \[ Z \otimes D(X) \to \iHom (X,Z) \] is an equivalence for every \(Z \in C\). Since both sides are exact functors in \(X\), it follows that the collection of objects \(X\) for which it is an equivalence forms a stable subcategory of \(C\). It is also closed under retracts since equivalences in \(C\) are closed under retracts. □
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