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Theorem 8.4.4.1. Let $\kappa $ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete. If $X \in \operatorname{\mathcal{S}}_{< \kappa }$ is a contractible Kan complex, then evaluation at $X$ induces an equivalence of $\infty $-categories
\[ \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{S}}_{< \kappa }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad T \mapsto T(X). \]
Proof.
Since $X$ is contractible, the identity functor $\operatorname{\mathcal{S}}_{< \kappa } \rightarrow \operatorname{\mathcal{S}}_{< \kappa }$ is corepresentable by $X$ (see Proposition 5.6.6.17), so that the inclusion functor
\[ \Delta ^0 \simeq \{ X\} \hookrightarrow \operatorname{\mathcal{S}}_{< \kappa } \simeq \operatorname{Fun}( (\Delta ^0)^{\operatorname{op}} , \operatorname{\mathcal{S}}_{< \kappa } ) \]
is a covariant Yoneda embedding for the standard $0$-simplex $\Delta ^0$. The desired result now follow from Theorem 8.4.3.2.
$\square$