Remark 6.2.4.3. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories which satisfies the equivalent conditions of Corollary 6.2.4.2, so that $G$ admits a left adjoint $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. For each object $X \in \operatorname{\mathcal{C}}$, the value $F(X) \in \operatorname{\mathcal{D}}$ admits several characterizations:
The object $F(X)$ corepresents the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor
\[ \mathrm{h} \mathit{\operatorname{\mathcal{D}}} \xrightarrow { \mathrm{h} \mathit{G} } \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \xrightarrow { \underline{\operatorname{Hom}}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( X, \bullet ) } \mathrm{h} \mathit{\operatorname{Kan}}. \]The object $F(X)$ corepresents the functor of $\infty $-categories
\[ \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{C}}\xrightarrow { h^{X} } \operatorname{\mathcal{S}}, \]where $h^ X$ is the functor corepresented by $X$.
The object $F(X)$ is the image in $\operatorname{\mathcal{D}}$ of an initial object of the $\infty $-category $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$.