Remark 8.1.10.10. In the situation of Theorem 8.1.10.9, the morphism
\[ \overline{V}: \operatorname{Cospan}^{\widetilde{L}, \widetilde{R}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) \]
is a locally cartesian fibration. To prove this, we observe that Remark 8.1.9.11 guarantees that $\overline{V}$ is an inner fibration of $(\infty ,2)$-categories and that the diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) ) \ar [d]^{V} \ar [r] & \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R} }( \operatorname{\mathcal{E}}) \ar [d]^{ \overline{V} } \\ \operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) ) \ar [r] & \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) } \]
is a pullback square. Since every morphism of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is contained in the pith $\operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$, the desired result follows from Theorem 8.1.10.9 (see Remark 5.1.5.6).