Proposition 6.1.4.1. 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 $\operatorname{\mathcal{C}}$. Assume that either $f$ or $g$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then $\eta $ is the unit of an adjunction (in the sense of Definition 6.1.2.10) if and only if it is an isomorphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C)$.
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Proof of Proposition 6.1.4.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms in $\operatorname{\mathcal{C}}$, and assume that $g$ is an isomorphism (the case where $f$ is an isomorphism can be treated by applying a similar argument in the opposite $2$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$). Suppose first that $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ is an isomorphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C)$. It follows that the horizontal compositions
\[ (\operatorname{id}_{f} \circ \eta ): f \Rightarrow f \circ (g \circ f) \quad \quad (\eta \circ \operatorname{id}_{g}): g \Rightarrow (g \circ f) \circ g \]
are isomorphisms in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)$ and $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(D,C)$, respectively. For each object $T \in \operatorname{\mathcal{C}}$, our assumption that $g$ is an isomorphism guarantees that the composition functor $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( T, D) \xrightarrow { g \circ } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,C)$ is an equivalence of categories, and therefore fully faithful. Invoking the criterion of Proposition 6.1.4.7, we conclude that $\eta $ is the unit of an adjunction.
We now prove the converse. Suppose that the $2$-morphism $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ is the unit of an adjunction. Our assumption that $g$ is an isomorphism guarantees that we can choose a $1$-morphism $f': C \rightarrow D$ and an isomorphism $\eta ': \operatorname{id}_{C} \xRightarrow {\sim } g \circ f'$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C)$. It follows from the first part of the proof that $\eta '$ is the unit of an adjunction. Applying Corollary 6.1.3.7, we deduce that there is a unique isomorphism $\beta : f \xRightarrow {\sim } f'$ for which $\eta '$ is equal to the composition
\[ \operatorname{id}_{C} \xRightarrow {\eta } g \circ f \xRightarrow {\operatorname{id}_ g \circ \beta } g \circ f'. \]
Since $\eta '$ and $(\operatorname{id}_ g \circ \beta )$ are isomorphisms in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C)$, it follows that $\eta $ is also an isomorphism. $\square$