Kerodon

Remark 6.1.6.3 (Adjoints as Duals). Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, let $f,g: X \rightarrow X$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{X} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{X}$ be $2$-morphisms of $\operatorname{\mathcal{C}}$. Then the pair $(\eta , \epsilon )$ is an adjunction in the $2$-category $\operatorname{\mathcal{C}}$ (in the sense of Definition 6.1.1.1) if and only if it is a duality datum in the monoidal category $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ of Remark 2.2.1.7.