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 $F$*.

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