Corollary 7.3.9.7. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a cocartesian fibration of $\infty $-categories, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that the following conditions are satisfied:
For every object $C \in \operatorname{\mathcal{C}}$ and every object $E \in \operatorname{\mathcal{E}}$, the $\infty $-category $\operatorname{\mathcal{D}}_{E} = \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ admits $\operatorname{\mathcal{C}}^{0}_{/C}$-indexed colimits.
For every object $C \in \operatorname{\mathcal{C}}$ and every morphism $e: E \rightarrow E'$ in $\operatorname{\mathcal{E}}$, the covariant transport functor $e_{!}: \operatorname{\mathcal{D}}_{E} \rightarrow \operatorname{\mathcal{D}}_{E'}$ preserves $\operatorname{\mathcal{C}}^{0}_{/C}$-indexed colimits.
Then every lifting problem
admits a solution $\overline{F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.