Definition 6.1.6.13. Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$. We will say that a morphism $\operatorname{ev}: X \otimes Y \rightarrow {\bf 1}$ exhibits $Y$ as a weak right dual of $X$ if, for every object $W \in \operatorname{\mathcal{C}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X \otimes W, X \otimes Y) \xrightarrow {\operatorname{ev}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X \otimes W, {\bf 1} ) \]
is bijective. We say that $\operatorname{ev}$ exhibits $X$ as a weak left dual of $Y$ if, for every object $Z \in \operatorname{\mathcal{C}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Z,X) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Z \otimes Y, X \otimes Y) \xrightarrow {\operatorname{ev}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Z \otimes Y, {\bf 1} ) \]
is bijective.