Remark 8.6.3.10. In the situation of Construction 8.6.3.9, the morphism $\Psi $ can be described explicitly on low-dimensional simplices as follows:
If $X$ is a vertex of $\operatorname{\mathcal{E}}^{\dagger }$ having image $C = U^{\dagger }(X)$, then $\Psi (X)$ is the vertex of $\operatorname{Cospan}( \operatorname{\mathcal{E}})$ corresponding to the vertex $T( X, \operatorname{id}_{ C } ) \in \operatorname{\mathcal{E}}$.
Let $X$ and $Y$ be vertices of $\operatorname{\mathcal{E}}^{\dagger }$, having images $C= U^{\dagger }(X)$ and $D = U^{\dagger }(Y)$. Let $f: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{E}}^{\dagger }$, and let us identify $U^{\dagger }(f)$ with an edge $e: D \rightarrow C$ in the simplicial set $\operatorname{\mathcal{C}}$. Then $\Psi ( f): \Psi ( X) \rightarrow \Psi ( Y)$ is the edge of $\operatorname{Cospan}(\operatorname{\mathcal{E}})$ corresponding to the pair of edges $T(X, \operatorname{id}_{C} ) \xrightarrow { T( f , e_{R} ) } T( Y, e ) \xleftarrow { T(\operatorname{id}_{Y}, e_{L} ) } T(Y, \operatorname{id}_{D} )$ in $\operatorname{\mathcal{E}}$; here $e_{L}: \operatorname{id}_{D} \rightarrow e$ and $e_{R}: \operatorname{id}_{C} \rightarrow e$ denote the edges of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ described in Example 8.1.3.6.