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 left 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.