Corollary 7.3.3.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty $-categories, and suppose that $\operatorname{\mathcal{C}}$ contains an initial object. The following conditions are equivalent:
- $(1)$
The functor $F$ is $U$-left Kan extended from the full subcategory $\operatorname{\mathcal{C}}^{\mathrm{init}} \subseteq \operatorname{\mathcal{C}}$ spanned by the initial objects.
- $(2)$
The functor $F$ carries every morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.