Definition 6.1.6.9. Let $\operatorname{\mathcal{C}}$ be a monoidal category. Then:
We say that an object $X \in \operatorname{\mathcal{C}}$ is right dualizable if there exists an object $Y \in \operatorname{\mathcal{C}}$ and a duality datum $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$. In this case, we will also say that $Y$ is a right dual of $X$, or that the morphism $\operatorname{ev}$ exhibits $Y$ as a right dual of $X$.
We say that an object $Y \in \operatorname{\mathcal{C}}$ is left dualizable if there exists an object $X \in \operatorname{\mathcal{C}}$ and a duality datum $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$. In this case, we will also say that $X$ is a left dual of $Y$, or that the morphism $\operatorname{ev}$ exhibits $X$ as a left dual of $Y$.