Definition 6.1.1.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $C$ and $D$ be objects of $\operatorname{\mathcal{C}}$, and let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms in $\operatorname{\mathcal{C}}$. An adjunction between $f$ and $g$ is a pair of $2$-morphisms $( \eta , \epsilon )$, where $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ is a morphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C)$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ is a morphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,D)$, which satisfy the following compatibility conditions:
- $(Z1)$
The composition
\[ f \xRightarrow [\sim ]{\rho _{f}^{-1}} f \circ \operatorname{id}_{C} \xRightarrow {\operatorname{id}_ f \circ \eta } f \circ (g \circ f) \xRightarrow [\sim ]{\alpha _{f,g,f}} (f \circ g) \circ f \xRightarrow {\epsilon \circ \operatorname{id}_ f} \operatorname{id}_{D} \circ f \xRightarrow [\sim ]{\lambda _{f}} f \]is the identity $2$-morphism from $f$ to itself. Here $\lambda _{f}$ and $\rho _{f}$ are the left and right unit constraints of the $2$-category $\operatorname{\mathcal{C}}$ (Construction 2.2.1.11) and $\alpha _{f,g,f}$ is the associativity constraint for the $2$-category $\operatorname{\mathcal{C}}$.
- $(Z2)$
The composition
\[ g \xRightarrow [\sim ]{\lambda _{g}^{-1}} \operatorname{id}_{C} \circ g \xRightarrow {\eta \circ \operatorname{id}_ g} (g \circ f) \circ g \xRightarrow [\sim ]{\alpha _{g,f,g}^{-1}} g \circ (f \circ g) \xRightarrow {\operatorname{id}_{g} \circ \epsilon } g \circ \operatorname{id}_ D \xRightarrow [\sim ]{\rho _{g}} g \]is the identity $2$-morphism from $g$ to itself.
If these conditions are satisfied, then we will refer to $\eta $ as the unit of the adjunction $(\eta , \epsilon )$ and to $\epsilon $ as the counit of the adjunction $(\eta , \epsilon )$. In this case, we say that $(\eta , \epsilon )$ exhibits $f$ as a left adjoint of $g$, and also that it exhibits $g$ as a right adjoint of $f$.