Proposition 6.1.3.1 (02D5)

Proposition 6.1.3.1. 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 the unit of an adjunction. Then:

$(1)$

For every $1$-morphism $f': C \rightarrow D$, the function

\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)}( f, f' ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C) }( \operatorname{id}_ C, g \circ f') \quad \quad \gamma \mapsto (\operatorname{id}_ g \circ \gamma )\eta \]

is a bijection.

$(2)$

For every $1$-morphism $g': D \rightarrow C$, the function

\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,C)}( g, g' ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C) }( \operatorname{id}_ C, g' \circ f) \quad \quad \beta \mapsto (\beta \circ \operatorname{id}_{f}) \eta \]

is a bijection.

Proof. Let $\rho _{f}: f \circ \operatorname{id}_{C} \xRightarrow {\sim } f$ be the right unit constraint. To prove $(1)$, we observe that the composition

\[ \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)}( f \circ \operatorname{id}_ C, f' ) \xrightarrow [\sim ]{\rho _ f^{-1}} \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)}( f, f' ) \rightarrow \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,C) }( \operatorname{id}_ C, g \circ f') \]

is given by the formation of right adjuncts (see Example 6.1.2.2 and Remark 6.1.2.4), and is therefore bijective by (Proposition 6.1.2.5). Assertion $(2)$ follows by a similar argument. $\square$