Corollary 8.1.3.8. The twisted arrow functor
\[ \operatorname{Tw}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad \operatorname{\mathcal{D}}\mapsto \operatorname{Tw}(\operatorname{\mathcal{D}}) \]
has a right adjoint, given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Cospan}(\operatorname{\mathcal{C}})$.