Variant 6.1.2.12. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: C \rightarrow D$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ be a $2$-morphism of $\operatorname{\mathcal{C}}$. Then $\eta $ is the unit of an adjunction if and only if the following condition is satisfied:
For every object $T \in \operatorname{\mathcal{C}}$ and every pair of morphisms $c: C \rightarrow T$ and $d: D \rightarrow T$, the $2$-morphism $\eta $ determines a bijection
\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,T)}( c \circ g, d ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,T)}( c, d \circ f ), \]carrying each $2$-morphism $\beta : c \circ g \Rightarrow d$ to the composition
\[ c \xRightarrow [\sim ]{\rho _{c}^{-1}} c \circ \operatorname{id}_{C} \xRightarrow {\operatorname{id}_ c \circ \eta } c \circ (g \circ f) \xRightarrow [\sim ]{\alpha _{c,g,f}} (c \circ g) \circ f \xRightarrow {\beta \circ \operatorname{id}_ f} d \circ f. \]