Example 6.1.6.10. Let $\operatorname{\mathcal{C}}$ be a monoidal category. We say that an object $X \in \operatorname{\mathcal{C}}$ is invertible if there exists an object $Y \in \operatorname{\mathcal{C}}$ such that the tensor products $Y \otimes X$ and $X \otimes Y$ are isomorphic to the unit object ${\bf 1}$. If this condition is satisfied, then any choice of isomorphism ${\bf 1} \simeq Y \otimes X$ is a duality datum (this is a special case of Proposition 6.1.4.1). In particular, the object $Y$ is a right dual of $X$. Similarly, $Y$ is a left dual of $X$.
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