Corollary 8.3.1.9. Let $\kappa $ be an uncountable cardinal, let $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between locally $\kappa $-small $\infty $-categories and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor which is corepresented by an object $C \in \operatorname{\mathcal{C}}$. Then a functor $\mathscr {G}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is a left Kan extension of $\mathscr {F}$ along $T$ if and only if it is corepresentable by the object $T(C) \in \operatorname{\mathcal{D}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume that $\mathscr {G}$ is corepresentable by $T(C)$; we will show that it is a left Kan extension of $\mathscr {F}$ along $T$ (the reverse implication follows immediately from Corollary 8.3.1.8). Fix a vertex $\eta \in \mathscr {F}(C)$ which exhibits $\mathscr {F}$ as corepresented by $C$. It follows from Proposition 8.3.1.3 that evaluation at $\eta $ induces a homotopy equivalence of Kan complexes
\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})}( \mathscr {F}, \mathscr {G} \circ T) \rightarrow \mathscr {G}( T(C) ). \]
We can therefore choose a natural transformation $\beta : \mathscr {F} \rightarrow \mathscr {G} \circ T$ which carries $\eta $ to a vertex which exhibits $\mathscr {G}$ as corepresented by $T(C)$. Applying Corollary 8.3.1.8, we see that $\beta $ exhibits $\mathscr {G}$ as a left Kan extension of $\mathscr {F}$ along $T$. $\square$