Kerodon

Remark 6.1.1.3. 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 $(\eta ^{\operatorname{op}}, \epsilon ^{\operatorname{op}})$ it is an adjunction between $g^{\operatorname{op}}$ and $f^{\operatorname{op}}$ in the opposite $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (Construction 2.2.3.1). Note that in this case, $g^{\operatorname{op}}$ is the left adjoint, while $f^{\operatorname{op}}$ is the right adjoint.