Corollary 8.4.4.2. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category, let $\operatorname{\mathcal{D}}$ be an $\infty $-category which is cocomplete and locally small, and let $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $F$ preserves small colimits.
- $(2)$
The functor $F$ admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$.
Proof.
Assume that $F$ preserves small colimits; we will show that it admits a right adjoint (the reverse implication follows from Corollary 7.1.3.21). Choose covariant Yoneda embeddings
\[ h_{\bullet }^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad h_{\bullet }^{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}), \]
set $f = F \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$, and let $G$ denote the composite functor
\[ \operatorname{\mathcal{D}}\xrightarrow { h_{\bullet }^{\operatorname{\mathcal{D}}} } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow { \circ f^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}). \]
It follows from Proposition 8.4.4.1 that $G$ admits a left adjoint $F': \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ such that $F' \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$ is isomorphic to $f = F \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$. Since the functor $F'$ also preserves small colimits (Corollary 7.1.3.21), Theorem 8.4.0.3 implies that it is isomorphic to $F$. It follows that $G$ is also a right adjoint of $F$.
$\square$