Example 3.3.3.11. Let $X = \Delta ^ n$ be a standard simplex. Then the comparison map $\psi _{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \rightarrow \operatorname{Sd}(X)$ of Construction 3.3.3.10, can be identified with the nerve of the functor $\operatorname{{\bf \Delta }}_{X} \rightarrow \operatorname{Chain}[n]$, which carries each morphism $\Delta ^{m} \rightarrow \Delta ^{n}$ to the image of the underlying map of linearly ordered sets $[m] \rightarrow [n]$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$