Exercise 7.1.5.20. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories, and let $C \in \operatorname{\mathcal{C}}$ be an object having image $D = U(C)$ in $\operatorname{\mathcal{D}}$. Show that $C$ is $U$-initial if and only if the following condition is satisfied:
- $(\ast )$
For every morphism $f: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the covariant transport functor $f_{!}: \operatorname{\mathcal{C}}_{D} \rightarrow \operatorname{\mathcal{C}}_{D'}$ carries $C$ to an initial object of the $\infty $-category $\operatorname{\mathcal{C}}_{D'}$.
For a more general statement, see Proposition 7.3.9.2.