Example 8.1.7.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, corresponding to a diagram
8.19
\begin{equation} \begin{gathered}\label{equation:left-toot-consequence} \xymatrix@R =40pt@C=20pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl] & \cdots & X_{n-1,n-1} \ar [dr] \ar [dl] & & X_{n,n} \ar [dl] \\ & \cdots \ar [dr] & & \cdots \ar [dr] \ar [dl] & & \cdots \ar [dl] & \\ & & X_{0,n-1} \ar [dr] & & X_{1,n} \ar [dl] & & \\ & & & X_{0,n}. & & & \\ } \end{gathered} \end{equation}
Let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which contain all identity morphisms and have the left two-out-of-three property. Then $\sigma $ belongs to $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if each of the rightward-pointing morphisms displayed in (8.19) belong to $L$, and each of the leftward-pointing morphisms displayed in (8.19) belong to $R$.