Kerodon

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

Corollary 6.2.4.4. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between ordinary categories. The following conditions are equivalent:

$(1)$

The functor $G$ admits a right adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

$(2)$

For every object $X \in \operatorname{\mathcal{C}}$, the set-valued functor

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Set}\quad \quad Z \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Z) ) \]

is corepresentable.