Kerodon

Remark 6.2.1.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural tranformation. Then $\eta $ is the unit of an adjunction between $F$ and $G$ if and only if the opposite natural transformation $\eta ^{\operatorname{op}}: G^{\operatorname{op}} \circ F^{\operatorname{op}} \rightarrow \operatorname{id}_{ \operatorname{\mathcal{C}}^{\operatorname{op}} }$ is the counit of an adjunction between the functors $G^{\operatorname{op}}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$. Note that in this case, $\eta ^{\operatorname{op}}$ exhibits $G^{\operatorname{op}}$ as the left adjoint of $F^{\operatorname{op}}$.