Remark 8.1.9.11. In the situation of Proposition 8.1.9.10, suppose that the collections $L$ and $R$ are closed under composition. Then $\widetilde{L}$ and $\widetilde{R}$ are also closed under composition (see Corollary 5.1.2.4). Applying Proposition 8.1.6.7, we deduce that the simplicial sets $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ and $\operatorname{Cospan}^{\widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}})$ are $(\infty ,2)$-categories. Moreover, it follows from the proof of Proposition 8.1.9.10 that for every pushout diagram $\sigma :$
in $\operatorname{\mathcal{E}}$ where $f$ belongs to $\widetilde{L}$ and $g$ belongs to $\widetilde{R}$, the image $U(\sigma )$ is a pushout diagram in $\operatorname{\mathcal{C}}$. Combining this observation with Corollary 8.1.6.8 and Proposition 8.1.4.2, we see that a $2$-simplex of $\operatorname{Cospan}^{\widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}})$ is thin if and only if its image in $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is thin. In particular:
The induced map $\overline{V}: \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is a functor of $(\infty ,2)$-categories.
The functor $\overline{V}$ is an inner fibration (since it is a pullback of the inner fibration $\operatorname{Cospan}^{\mathrm{all}, \widetilde{R} }( \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\mathrm{all}, R}( \operatorname{\mathcal{C}})$ of Proposition 8.1.9.9).
The underlying functor $V: \operatorname{Pith}( \operatorname{Cospan}^{ \widetilde{L}, \widetilde{R}}( \operatorname{\mathcal{E}}) ) \rightarrow \operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$ is also an inner fibration (since it is a pullback of $\overline{V}$).