Remark 8.1.3.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets. Using Proposition 8.1.3.7, we obtain bijections
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Cospan}(\operatorname{\mathcal{D}}) ) & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n \times \operatorname{\mathcal{C}}, \operatorname{Cospan}(\operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}( \Delta ^ n \times \operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\Delta ^ n) \times \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\Delta ^ n), \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{Cospan}( \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) ) ). \end{eqnarray*}
These bijections depend functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore determine an isomorphism of simplicial sets $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Cospan}(\operatorname{\mathcal{D}}) ) \simeq \operatorname{Cospan}( \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) )$.