Kerodon

Remark 8.1.6.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $L$ and $R$ be collections of edges of $\operatorname{\mathcal{C}}$. Suppose we are given a morphism of simplicial sets $f: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$, corresponding to a morphism $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ (see Proposition 8.1.3.7). For every edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}$, let $e_{L}: \operatorname{id}_{X} \rightarrow e$ and $e_{R}: \operatorname{id}_{Y} \rightarrow e$ be the edges of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ described in Example 8.1.3.6. Then $f$ factors through the simplicial subset $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if the edge $F(e_{L})$ belongs to $L$ and the edge $F(e_{R} )$ belongs to $R$, for every edge $e$ of $\operatorname{\mathcal{D}}$.