Remark 6.1.6.14. Let $\operatorname{\mathcal{C}}$ be a monoidal category and let $X$ be an object of $\operatorname{\mathcal{C}}$. It follows immediately from the definition that if there exists a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ which exhibits $Y$ as a weak right dual of $X$, then the pair $(Y, \operatorname{ev})$ is unique up to isomorphism and depends functorially on $X$. To emphasize this dependence we will sometimes denote the object $Y$ by $X^{\vee }$ and abuse terminology by referring to it as the weak right dual of $X$.
Similarly, if $Y$ is a fixed object of $\operatorname{\mathcal{C}}$ and there exists a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ which exhibits $X$ as a weak left dual of $Y$, then the pair $(X,\operatorname{ev})$ is uniquely determined up to isomorphism and depends functorially on $Y$. We will emphasize this dependence by denoting the object $X$ by ${^{\vee }Y}$ and referring to it as the weak left dual of $Y$.