Kerodon

Construction 8.1.3.5. Let $\operatorname{\mathcal{D}}$ be a simplicial set and let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$ be an $n$-simplex of $\operatorname{\mathcal{D}}$. Invoking the functoriality of the twisted arrow construction, we obtain a map $\operatorname{Tw}( \Delta ^{n} ) \xrightarrow { \operatorname{Tw}(\sigma ) } \operatorname{Tw}(\operatorname{\mathcal{D}})$, which we can identify with an $n$-simplex $u( \sigma )$ of the simplicial set $\operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$. The construction $\sigma \mapsto u(\sigma )$ is compatible with face and degeneracy operators, and therefore determines a morphism of simplicial sets $u: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$ which we will refer to as the unit map.