$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 8.1.3.7. Let $\operatorname{\mathcal{D}}$ be a simplicial set and let $u: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$ be the unit map of Construction 8.1.3.5. For every simplicial set $\operatorname{\mathcal{C}}$, the composite map
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) & \rightarrow & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) ), \operatorname{Cospan}(\operatorname{\mathcal{C}}) ) \\ & \xrightarrow { \circ u} & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{D}}, \operatorname{Cospan}(\operatorname{\mathcal{C}}) ) \end{eqnarray*}
is a bijection.
Proof.
Let us regard the simplicial set $\operatorname{\mathcal{C}}$ as fixed. For every simplicial set $\operatorname{\mathcal{D}}$, the unit map $u$ of Construction 8.1.3.5 determines a function
\[ \theta _{\operatorname{\mathcal{D}}}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{D}}, \operatorname{Cospan}(\operatorname{\mathcal{C}}) ). \]
Using Remark 8.1.1.4, we see that the construction $\operatorname{\mathcal{D}}\mapsto \theta _{\operatorname{\mathcal{D}}}$ carries colimits (in the category of simplicial sets) to limits (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set})$). Consequently, to show that $\theta _{\operatorname{\mathcal{D}}}$ is a bijection, we may assume without loss of generality that $\operatorname{\mathcal{D}}= \Delta ^ n$ is a standard simplex (see Remark 1.1.3.13). In this case, the desired result follows immediately from the definition of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.
$\square$