Kerodon

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

Corollary 6.2.4.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. The following conditions are equivalent:

$(1)$

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

$(2)$

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

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}\quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(C), D) \]

is representable by an object of $\operatorname{\mathcal{C}}$.