Corollary 8.2.1.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \{ C\} $ has an initial object. Then the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$ has an initial object. Moreover, an object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$ is initial if and only if it satisfies the following pair of conditions:
- $(1)$
For every object $C \in \operatorname{\mathcal{C}}$, the image $F( \operatorname{id}_ C )$ is an initial object of the $\infty $-category $\operatorname{\mathcal{E}}_ C$.
- $(2)$
Let $e$ be a morphism of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ whose image in $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is an isomorphism. Then $F(e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$.