Remark 8.6.3.4 (Base Change). Suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r]^-{F} & \operatorname{\mathcal{C}}, } \]
where $U$ and $U'$ are cocartesian fibrations. Then we have a canonical isomorphism of simplicial sets
\[ \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}') \simeq \operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{\mathcal{C}}'^{\operatorname{op}}. \]
In particular, for each vertex $C \in \operatorname{\mathcal{C}}$, the fiber $\operatorname{Cospan}_{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \{ C\} $ is isomorphic to the $\infty $-category $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{E}}_{C} )$, which is equivalent to the $\infty $-category $\operatorname{\mathcal{E}}_{C}$.