Construction 3.3.3.10. Let $X$ be a simplicial set and let $\operatorname{{\bf \Delta }}_{X}$ denote the category of simplices of $X$ (Construction 1.1.3.9). Unwinding the definitions, we see that $n$-simplices $\sigma $ of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} )$ can be identified with diagrams of simplicial sets
For $0 \leq i \leq n$, let $S_{i} \subseteq [k_ n ]$ be the image of the underlying map of linearly ordered sets $[k_0] \rightarrow [k_ n]$, and suppose we are given a morphism $u: X \rightarrow \operatorname{Ex}(Y)$ which exhibits $Y$ as a subdivision of $X$. Then $u$ carries $\tau $ to a $k_{n}$-simplex of $\operatorname{Ex}(Y)$, which we can identify with a morphism $\operatorname{N}_{\bullet }( \operatorname{Chain}[k_ n] ) \rightarrow Y$ carrying $( S_0 \subseteq S_1 \subseteq \cdots \subseteq S_ n )$ to an $n$-simplex $\sigma '$ of $Y$. The construction $\sigma \mapsto \sigma '$ depends functorially on $[n]$, and therefore determines a comparison map $\psi _{X}: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{X} ) \rightarrow Y = \operatorname{Sd}(X)$.