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Variant 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:


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.


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