Remark 2.4.4.11. Let $S_{\bullet }$ be a simplicial set. Then the path category $\operatorname{Path}[S]_{\bullet }$ is characterized (up to isomorphism) by properties $(1)$, $(2)$, and $(3)$ of Theorem 2.4.4.10. More precisely, suppose that $\operatorname{\mathcal{C}}_{\bullet }$ is a simplicial category and that we are given a comparison map $u': S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ satisfying the following three conditions:
- $(1')$
The map $u'$ induces a bijection from the set of vertices of $S_{\bullet }$ to the set of objects of $\operatorname{\mathcal{C}}_{\bullet }$.
- $(2')$
For each nonnegative integer $m \geq 0$, the category $\operatorname{\mathcal{C}}_{m}$ is free.
- $(3')$
For each nonnegative integer $m \geq 0$, the construction $(\sigma , \overrightarrow {I} ) \mapsto u'(\sigma , \overrightarrow {I} )$ induces a bijection from $E(S,m)$ to the set of indecomposable morphisms of the category $\operatorname{\mathcal{C}}_ m$.
Then $u'$ exhibits $\operatorname{\mathcal{C}}_{\bullet }$ as a path category of $S_{\bullet }$, in the sense of Definition 2.4.4.1. To prove this, we invoke the universal property of $\operatorname{Path}[S]_{\bullet }$ to deduce that there is a unique simplicial functor $F: \operatorname{Path}[S]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ for which the composite map
is equal to $u'$. Combining Theorem 2.4.4.10 with assumptions $(1')$, $(2')$, and $(3')$, we deduce that for each $m \geq 0$, the induced functor $\operatorname{Path}[S]_{m} \rightarrow \operatorname{\mathcal{C}}_{m}$ is a map between free categories which is bijective on objects and indecomposable morphisms, and is therefore an isomorphism of categories.