Remark 6.1.6.2 (Duals as Adjoints). Let $\operatorname{\mathcal{C}}$ be a monoidal category containing objects $X$ and $Y$, which we regard as $1$-morphisms of the $2$-category $B\operatorname{\mathcal{C}}$ described in Example 2.2.2.5. Suppose we are given a pair of morphisms
\[ \operatorname{coev}: {\bf 1} \rightarrow Y \otimes X \quad \quad \operatorname{ev}: X \otimes Y \rightarrow {\bf 1} \]
in $\operatorname{\mathcal{C}}$, which we identify with $2$-morphisms of $B\operatorname{\mathcal{C}}$. Then the pair $(\operatorname{coev}, \operatorname{ev})$ is a duality datum in the monoidal category $\operatorname{\mathcal{C}}$ (in the sense of Definition 6.1.6.1) if and only if it is an adjunction in the $2$-category $B\operatorname{\mathcal{C}}$ (in the sense of Definition 6.1.1.1).