Kerodon

Definition 6.2.1.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. We say that $F$ is a left adjoint of $G$, or that $G$ is a right adjoint of $F$, if there exists a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which is the unit of an adjunction between $F$ and $G$. In this case, we say that $\eta $ exhibits $F$ as a left adjoint of $G$ and also that it exhibits $G$ as a right adjoint of $F$. Equivalently, $F$ is a left adjoint of $G$ if there exists a natural transformation $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is the counit of an adjunction between $F$ and $G$; in this case, we say that $\epsilon $ exhibits $F$ as a left adjoint of $G$ and also that it exhibits $G$ as a right adjoint of $F$.