Proposition 6.1.2.13. 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 $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ 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 left adjuncts with respect to $\epsilon $ (Construction 6.1.2.1) induces a bijection
\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,C)}( c, g \circ d) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,D)}( f \circ c, d ) \]- $(2)$
For every object $T \in \operatorname{\mathcal{C}}$ and every pair of $1$-morphisms $c: C \rightarrow T$ and $d: D \rightarrow T$, the $2$-morphism $\epsilon $ determines a bijection
\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,T)}( c, d \circ f ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,T)}( c \circ g, d ) \]carrying each $2$-morphism $\gamma : c \Rightarrow d \circ f$ to the composition
\[ c \circ g \xRightarrow {\gamma \circ \operatorname{id}_ g} (d \circ f) \circ g \xRightarrow [\sim ]{\alpha _{d,f,g}^{-1}} d \circ (f \circ g) \xRightarrow {\operatorname{id}_{d} \circ \epsilon } d \circ \operatorname{id}_{D} \xRightarrow [\sim ]{\rho _{d}} d. \]- $(3)$
There exists a $2$-morphism $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ for which $(\eta , \epsilon )$ is an adjunction between $f$ and $g$.
Moreover, if these conditions are satisfied, then the $2$-morphism $\eta $ is uniquely determined.