Proposition 8.3.1.3 ($\infty $-Categorical Yoneda Lemma). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor, and let $\eta \in \mathscr {F}(X)$ be a vertex which exhibits the functor $\mathscr {F}$ as corepresented by $X$ (see Definition 5.6.6.1). Then, for every functor $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, the comparison map
\[ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {F}, \mathscr {G} ) \rightarrow \mathscr {G}(X) \]
of Notation 8.3.1.2 is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.