$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 6.1.2.9. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ be a $2$-morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
For every object $T \in \operatorname{\mathcal{C}}$ and every pair of $1$-morphisms $c: T \rightarrow C$ and $d: T \rightarrow D$, the formation of right adjuncts with respect to $\eta $ (Construction 6.1.2.1) induces a bijection
\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,D)}( f \circ c, d ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,C)}( c, g \circ d). \]
- $(2)$
There exists a $2$-morphism $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ for which $(\eta , \epsilon )$ is an adjunction between $f$ and $g$.
Moreover, if these conditions are satisfied, then the $2$-morphism $\epsilon $ is uniquely determined.
Proof.
The implication $(2) \Rightarrow (1)$ follows from Corollary 6.1.2.6. Conversely, suppose that condition $(1)$ is satisfied. Applying $(1)$ in the case $T = D$, $c = g$, and $d = \operatorname{id}_{D}$, we conclude that there is a unique $2$-morphism $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ whose right adjunct is equal to the inverse $\rho _{g}^{-1}: g \xRightarrow {\sim } g \circ \operatorname{id}_{D}$ of the right unit constraint $\rho _ g$, so that the pair $(\eta , \epsilon )$ satisfies condition $(Z2)$ of Definition 6.1.1.1 (Example 6.1.2.3). We will complete the proof by showing that $(\eta , \epsilon )$ also satisfies condition $(Z1)$. Let $\gamma : f \circ \operatorname{id}_{C} \Rightarrow f$ be the left adjunct of $\eta $. It follows from Proposition 6.1.2.5 that the right adjunct of $\gamma $ is equal to $\eta $, which is also the right adjunct of the unit constraint $\rho _{f}: f \circ \operatorname{id}_{C} \xRightarrow {\sim } f$. Invoking assumption $(1)$, we conclude that $\gamma = \rho _{f}$, which is a restatement of $(Z1)$ (Example 6.1.2.3).
$\square$