$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 8.4.4.3. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category and let $\mathscr {F}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $\mathscr {F}$ admits a left adjoint.
- $(2)$
The functor $\mathscr {F}$ is representable by an object of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$.
- $(3)$
The functor $\mathscr {F}$ preserves small limits.
Proof.
Since the identity functor $\operatorname{id}: \operatorname{\mathcal{S}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable (by the object $\Delta ^0 \in \operatorname{\mathcal{S}}$), the implication $(1) \Rightarrow (2)$ follows from Corollary 6.2.4.2. The implication $(2) \Rightarrow (3)$ is a special case of Corollary 7.4.5.17, and the implication $(3) \Rightarrow (1)$ follows by applying Corollary 8.4.4.2 to to the opposite functor $\mathscr {F}^{\operatorname{op}}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}^{\operatorname{op}}$.
$\square$