Remark 7.3.3.24. In the situation of Corollary 7.3.3.23, the functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if, for every morphism $f: E \rightarrow E'$ in the $\infty $-category $\operatorname{\mathcal{E}}$, the composite map
\[ \operatorname{\mathcal{C}}\xrightarrow {F} \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\hookrightarrow \Delta ^{1} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}} \]
is left Kan extended from the full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. See Remark 7.1.6.25.