Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.3.7.8. Let $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ be full subcategories. Then $\overline{F}$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it satisfies the following pair of conditions:

$(1)$

The functor $\overline{F}$ is left Kan extended from $\operatorname{\mathcal{C}}$.

$(2)$

The restriction $\overline{F}|_{\operatorname{\mathcal{C}}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proof. Apply Proposition 7.3.7.6 in the special case $\operatorname{\mathcal{E}}= \Delta ^0$. $\square$