Remark 6.1.6.17. Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$. If both $X$ and $Y$ are right dualizable, then the tensor product $X \otimes Y$ is also right dualizable; moreover we have a canonical isomorphism $(X \otimes Y)^{\vee } \simeq Y^{\vee } \otimes X^{\vee }$ (see Corollary 6.1.5.4 for a more precise statement). Similarly, if both $X$ and $Y$ are left dualizable, then the tensor product $X \otimes Y$ is left dualizable, and there is a canonical isomorphism ${^{\vee }(X \otimes Y)} \simeq {^{\vee }Y} \otimes {^{\vee }X}$.
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