Definition 8.1.6.1 (Restricted Cospans). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $L$ and $R$ be collections of edges of $\operatorname{\mathcal{C}}$. We let $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ denote the simplicial subset of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ whose $n$-simplices are given by diagrams $X: \operatorname{Tw}(\Delta ^ n) \rightarrow \operatorname{\mathcal{C}}$ which satisfy the following condition:
For every pair of integers $0 \leq i \leq j \leq n$, the edge $X_{i,i} \rightarrow X_{i,j}$ belongs to $L$ and the edge $X_{j,j} \rightarrow X_{i,j}$ belongs to $R$.