$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 6.1.3.3. 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 $(\eta , \epsilon )$ be an adjunction between $f$ and $g$. Let $g': D \rightarrow C$ be another $1$-morphism of $\operatorname{\mathcal{C}}$. Then:
- $(1)$
For every $2$-morphism $\eta ': \operatorname{id}_{C} \Rightarrow g' \circ f$, there is a unique $2$-morphism $\beta : g \Rightarrow g'$ for which $\eta '$ is equal to the composition $\operatorname{id}_{C} \xRightarrow {\eta } g \circ f \xRightarrow {\beta \circ \operatorname{id}_ f} g' \circ f$. Moreover, $\beta $ is an isomorphism if and only if $\eta '$ is the unit of an adjunction.
- $(2)$
For every $2$-morphism $\epsilon ': f \circ g' \Rightarrow \operatorname{id}_ D$, there is a unique $2$-morphism $\gamma : g' \Rightarrow g$ for which $\epsilon '$ factors as a composition $f \circ g' \xRightarrow {\operatorname{id}_ f \circ \gamma } f \circ g \xRightarrow {\epsilon } \operatorname{id}_ D$. Moreover, $\gamma $ is an isomorphism if and only $\epsilon '$ is the counit of an adjunction.
Proof.
We will prove $(1)$; the proof of $(2)$ similar. Let $\eta ': \operatorname{id}_{C} \Rightarrow g' \circ f$ be a $2$-morphism of $\operatorname{\mathcal{C}}$. It follows from Proposition 6.1.3.1 that there is a unique $2$-morphism $\beta : g \Rightarrow g'$ satisfying $\eta ' = (\beta \circ \operatorname{id}_ f)\eta $. If $\beta $ is an isomorphism, then $\eta '$ is the unit of an adjunction by virtue of Remark 6.1.1.5. Conversely, suppose that $\eta '$ is the unit of an adjunction. To prove that $\beta $ is an isomorphism, it will suffice to show that for every $1$-morphism $g'': D \rightarrow C$, precomposition with $\beta $ induces a bijection $\operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C) }( g', g'' ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C) }( g, g'')$. This is clear: we have a commutative diagram
\[ \xymatrix@R =50pt@C=30pt{ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C) }( g', g'' ) \ar [rr]^{ \beta } \ar [dr]_{\eta '} & & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C) }( g, g'') \ar [dl]^{\eta } \\ & \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C) }( \operatorname{id}_{C}, g'' \circ f ), & } \]
where the vertical maps are bijective by virtue of Proposition 6.1.3.1.
$\square$