Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Warning 6.1.3.6. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f: C \rightarrow D$ be a $1$-morphism of $\operatorname{\mathcal{C}}$. It follows from Corollary 6.1.3.3 that if $f$ admits a right adjoint $f^ R$, then $f^ R$ is characterized (up to canonical isomorphism) by the requirement that it represents the functor

\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(C,D)^{\operatorname{op}} \rightarrow \operatorname{Set}\quad \quad g \mapsto \operatorname{Hom}_{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(D,D) }( f \circ g, \operatorname{id}_ D). \]

Beware that it is possible for this functor to be representable by a $1$-morphism $g: D \rightarrow C$ which is not a right adjoint to $f$ (in which case $f$ cannot admit any right adjoint); see Warning 6.1.6.16.