$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Let $(Q, \leq )$ be a partially ordered set, which we regard as a category (having a unique morphism from $x$ to $y$ when $x \leq y$). Note that, for every pair of elements $x,y \in Q$, the category $\underline{\operatorname{Hom}}_{\operatorname{Path}[Q]_{(2)}}(x,y)$ is empty unless $x \leq y$. It follows that there is a unique (strict) functor $\operatorname{Path}[Q]_{(2)} \rightarrow Q$ which is the identity on objects.