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Definition Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$. We say that a $2$-morphism $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ is the unit of an adjunction if it satisfies the equivalent conditions of Proposition that is, if there exists a $2$-morphism $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ for which the pair $(\eta , \epsilon )$ is an adjunction. If this condition is satisfied, we will say that $\eta $ exhibits $f$ as a left adjoint of $g$ and also that $\eta $ exhibits $g$ as a right adjoint of $f$.