Kerodon

Remark 8.1.6.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $L$ and $R$ be collections of edges of $\operatorname{\mathcal{C}}$, and let $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ denote the restricted cospan construction of Definition 8.1.6.1. Note that a morphism of simplicial sets $K \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ factors through $\operatorname{Cospan}^{L,R}( \operatorname{\mathcal{C}})$ if and only if its restriction to the $1$-skeleton $\operatorname{sk}_1(K)$ factors through $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$. In particular, if $\sigma $ is a $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ which is thin when viewed as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, then it is also thin when viewed as a $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$. For a partial converse, see Corollary 8.1.6.8.