Lemma 11.2.1. Let \(C\) be a symmetric monoidal \(\infty \)-category with filtered colimits, and assume that the tensor product in \(C\) preserves filtered colimits in both variables. If the monoidal unit \(\unit \in C\) is compact, then every dualizable object is compact.
Proof. Let \(X\) be a dualizable object in \(C\), and let \(X^{\vee }\) denote its dual. Then the functor \(X \otimes -\colon C \to C\) admits a right adjoint of the form \(X^{\vee } \otimes -\colon C \to C\). In particular, we have \[ \Hom _C(X,-) \simeq \Hom _C(X \otimes \unit ,-) \simeq \Hom _C(\unit ,X^{\vee } \otimes -), \] and the right-hand side preserves filtered colimits by the assumptions on \(C\). □
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