Kerodon

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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 $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$.