Construction 11.9.1.3. Let $Q$ be a partially ordered set and let $Q^{\operatorname{op}}$ denote the opposite partially ordered set. To avoid confusion, for each element $q \in Q$, we write $q^{\operatorname{op}}$ for the corresponding element of $Q^{\operatorname{op}}$. Let $\operatorname{Tw}(Q)$ denote the twisted arrow category of $Q$ (Example 8.1.0.5), which we identify with the partially ordered subset of $Q^{\operatorname{op}} \times Q$ consisting of those pairs $(p^{\operatorname{op}}, q)$ satisfying $p \leq q$. We then have a morphism of partially ordered sets $\xi _{Q}: \operatorname{Tw}(Q) \times [1] \rightarrow Q^{\operatorname{op}} \star Q$, given concretely by the formulae
\[ \xi _{Q}( p^{\operatorname{op}}, q, i ) = \begin{cases} p^{\operatorname{op}} & \text{ if } i = 0 \\ q & \text{ if } i = 1. \end{cases} \]
Let $\operatorname{\mathcal{C}}$ be a simplicial set. For every nonempty finite linearly ordered set $Q$, we obtain a map
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(Q), \operatorname{Tw}(\operatorname{\mathcal{C}}) ) & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(Q^{\operatorname{op}} \star Q), \operatorname{\mathcal{C}}) \\ & \xrightarrow { \circ \xi _{Q}} & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{Tw}(Q) \times [1]), \operatorname{\mathcal{C}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}( \operatorname{N}_{\bullet }(Q) ) \times \Delta ^1, \operatorname{\mathcal{C}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}( \operatorname{N}_{\bullet }(Q) ), \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }(Q), \operatorname{Cospan}( \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) ) ). \end{eqnarray*}
This construction depends functorially on $Q$, and therefore determines a morphism of simplicial sets $\Xi : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) )$.