Lemma 8.6.5.13 (04E5)

Lemma 8.6.5.13. Let $\kappa $ be an uncountable cardinal, let $\operatorname{\mathcal{E}}$ be an $\infty $-category which is locally $\kappa $-small, and let $\mathscr {F}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor. Suppose we are given a full subcategory $\operatorname{\mathcal{E}}_{0} \subseteq \operatorname{\mathcal{E}}$, an object $X \in \operatorname{\mathcal{E}}_0$, and a vertex $\eta \in \mathscr {F}(X)$. Then $\eta $ exhibits the $\mathscr {F}$ as corepresented by the object $X$ if and only if the following conditions are satisfied:

$(a)$: The vertex $\eta $ exhibits $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{E}}_0}$ as corepresented by the object $X$.
$(b)$: The functor $\mathscr {F}$ is left Kan extended from $\operatorname{\mathcal{E}}_0$.

Proof. It follows immediately from the definition that if $\eta $ exhibits $\mathscr {F}$ as corepresented by $X$, then it also exhibits $\mathscr {F}_0$ as corepresented by $X$. We may therefore assume without loss of generality that condition $(a)$ is satisfied. In this case, the desired equivalence follows immediately by combining the criterion of Lemma 8.3.1.7 with the transitivity property of Kan extensions (Proposition 7.3.8.18). $\square$