Kerodon

Remark 6.1.1.4. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ be $2$-morphisms of $\operatorname{\mathcal{C}}$. Then the pair $(\eta , \epsilon )$ is an adjunction between $f$ and $g$ in the $2$-category $\operatorname{\mathcal{C}}$ if and only if the pair $(\epsilon ^{\operatorname{c}}, \eta ^{\operatorname{c}})$ is an adjunction between $g^{\operatorname{c}}$ and $f^{\operatorname{c}}$ in the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ (Construction 2.2.3.4). Note that in this case, $\epsilon ^{\operatorname{c}}$ is the unit of the adjunction and $\eta ^{\operatorname{c}}$ is the counit. Similarly, $g^{\operatorname{c}}$ is the left adjoint and $f^{\operatorname{c}}$ is the right adjoint.