Proposition 1.2.3.15. Let $\operatorname{\mathcal{C}}$ be a category, let $Q^{\bullet }$ be a cosimplicial object of $\operatorname{\mathcal{C}}$, and let $\operatorname{Sing}^{Q}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the functor of Variant 1.2.2.8. If the category $\operatorname{\mathcal{C}}$ admits small colimits, then the functor $\operatorname{Sing}^{Q}_{\bullet }$ admits a left adjoint $\operatorname{Set_{\Delta }}\rightarrow \operatorname{\mathcal{C}}$, which we will denote by $S \mapsto | S |^{Q}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $S$ be a simplicial set; we wish to show that the functor
\[ \lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}\quad \quad C \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{Sing}^{Q}_{\bullet }(C) ) \]
is corepresentable by an object $|S|^{Q} \in \operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ admits small colimits, the collection of corepresentable functors from $\operatorname{\mathcal{C}}$ to $\operatorname{Set}$ is closed under the formation of small limits. Using Remark 1.1.3.13 (or Lemma 1.2.3.13), we can reduce to the case where $S = \Delta ^{n}$ is a standard simplex. In this case, the functor $\lambda $ is corepresented by the object $Q^{n} \in \operatorname{\mathcal{C}}$ (see Proposition 1.1.0.12). $\square$