Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Variant 6.1.3.2. 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 the counit of an adjunction. Then:

$(1)$

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

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

is a bijection.

$(2)$

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

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

is a bijection.

Proof. Apply Proposition 6.1.3.1 to the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$. $\square$