Proposition 8.4.4.1. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is essentially small, that $\operatorname{\mathcal{D}}$ is cocomplete and locally small, and let
\[ h_{\bullet }^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad h_{\bullet }^{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \]
be covariant Yoneda embeddings for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. Let $G$ denote the composite functor
\[ \operatorname{\mathcal{D}}\xrightarrow { h_{\bullet }^{\operatorname{\mathcal{D}}} } \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \xrightarrow { \circ f^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}). \]
Then the functor $G$ admits a left adjoint $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$. Moreover, the composition $F \circ h_{\bullet }^{\operatorname{\mathcal{C}}}$ is isomorphic to $f$.