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Notation Let $S_{\bullet }$ be a simplicial set. For each nonnegative integer $m$, we let $E(S,m)$ denote the collection of pairs $(\sigma , \overrightarrow {I} )$, where $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$ is a nondegenerate simplex of $S_{\bullet }$ of dimension $n > 0$ and $\overrightarrow {I} = (I_0 \supseteq I_{1} \supseteq \cdots \supseteq I_{m-1} \supseteq I_ m )$ is a chain of subsets of $[n]$ satisfying $I_0 = [n]$ and $I_ m = \{ 0, n \} $. Here we will view $\overrightarrow {I}$ as a $m$-simplex of the simplicial set $\operatorname{Hom}_{ \operatorname{Path}[n] }(0, n)_{\bullet }$.

Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ be a morphism of simplicial sets. For each element $(\sigma , \overrightarrow {I}) \in E(S,m)$, the composite map

\[ \Delta ^{n} \xrightarrow {\sigma } S_{\bullet } \xrightarrow {u} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \]

can be identified with a simplicial functor $u(\sigma ): \operatorname{Path}[n] \rightarrow \operatorname{\mathcal{C}}$. This functor carries $\overrightarrow {I}$ to a morphism in the ordinary category $\operatorname{\mathcal{C}}_{m}$, which we will denote by $u(\sigma , \overrightarrow {I} )$.