Example 6.1.2.3. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing $1$-morphisms $f: C \rightarrow D$ and $g: D \rightarrow C$, and suppose we are given $2$-morphisms $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$. Then $(\eta , \epsilon )$ is an adjunction between $f$ and $g$ if and only if the following conditions are satisfied:
- $(Z1)$
The left adjunct of $\eta $ (with respect to $\epsilon $) is equal to the right unit constraint $\rho _{f}: f \circ \operatorname{id}_{C} \xRightarrow {\sim } f$.
- $(Z2)$
The right adjunct of $\epsilon $ (with respect to $\eta $) is the inverse $\rho _{g}^{-1}: g \xRightarrow {\sim } g \circ \operatorname{id}_{D}$ of the right unit constraint.