$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 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 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 $S_{\bullet } \mapsto | S_{\bullet } |^{Q}$.

Proof. Let us say that a simplicial set $S_{\bullet }$ is good if the functor

\[ (C \in \operatorname{\mathcal{C}}) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{Sing}^{Q}_{\bullet }(C) ) \]

is corepresentable by an object of the category $\operatorname{\mathcal{C}}$ (in which case we denote the corepresenting object by $|S_{\bullet }|^{Q}$). It follows from Yoneda's lemma that the standard $n$-simplex $\Delta ^ n$ is good for each $n \geq 0$, with $| \Delta ^{n} |^{Q} \simeq Q^{n}$. If $\operatorname{\mathcal{C}}$ admits small colimits, then the proof of Lemma shows that the collection of good simplicial sets is closed under small colimits. It now suffices to observe that every simplicial set $S_{\bullet }$ can be written as a small colimit of simplices (Lemma $\square$