Variant 3.3.3.20. The proof of Lemma 3.3.3.19 also shows that the functor
\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad X \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \]
preserves colimits. Consequently, the comparison maps $\psi _{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \rightarrow \operatorname{Sd}(X)$ of Construction 3.3.3.10 are uniquely determined by the following properties:
The construction $X \mapsto \psi _{X}$ is functorial: that is, it determines a natural transformation from the functor $X \mapsto \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} )$ to the subdivision functor $\operatorname{Sd}$.
When $X = \Delta ^ n$ is a standard simplex, $\psi _{X}$ is the nerve of the functor
\[ \operatorname{{\bf \Delta }}_{X} \rightarrow \operatorname{Chain}[n] \quad \quad (\alpha : [m] \rightarrow [n] \mapsto \mathrm{im}(\alpha ) \subseteq [n] ) \]described in Example 3.3.3.11.