# Kerodon

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Example 8.4.0.4. Let $X$ be a contractible Kan complex, which we identify with a vertex $x$ of the simplicial set $\operatorname{\mathcal{S}}$. Applying Theorem 8.4.0.3 in the special case where $\operatorname{\mathcal{C}}= \Delta ^0$, we deduce that the map $x: \Delta ^0 \rightarrow \operatorname{\mathcal{S}}$ exhibits $\operatorname{\mathcal{S}}$ as a cocompletion of the $0$-simplex $\Delta ^0$. That is, for every $\infty$-category $\operatorname{\mathcal{D}}$ which admits small colimits, the evaluation map

$\operatorname{Fun}'( \operatorname{\mathcal{S}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}\quad \quad F \mapsto F( X )$

is an equivalence of $\infty$-categories, where $\operatorname{Fun}'( \operatorname{\mathcal{S}}, \operatorname{\mathcal{D}})$ denotes the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{S}}, \operatorname{\mathcal{D}})$ spanned by the colimit-preserving functors. Note that this property characterizes the $\infty$-category $\operatorname{\mathcal{S}}$ up to equivalence: it is “freely generated” under small colimits by the object $\Delta ^0$.