Warning 8.6.3.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Proposition 8.6.3.5 guarantees the existence of a morphism
which exhibits the projection map $\operatorname{Cospan}^{\dagger }(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ as a cartesian conjugate of $U$. Beware that the construction of $T$ requires making some auxiliary choices. For example, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then we can construct the datum $T$ by choosing a homotopy inverse to the equivalence $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Cospan}^{\dagger }( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}})$ of Example 8.6.3.12.