$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.3.2.13. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$, $F_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$, and $F_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories. The following conditions are equivalent:
- $(1)$
There exists a functor $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left Kan extended from $\operatorname{\mathcal{K}}$ which satisfies $F_0 = F|_{\operatorname{\mathcal{K}}}$ and $F_1 = F|_{\operatorname{\mathcal{C}}}$.
- $(2)$
There exists a natural transformation $\beta : F_0 \rightarrow F_1 \circ \delta $ which exhibits $F_1$ as a left Kan extension of $F_0$ along $\delta $.
Proof.
The implication $(1) \Leftrightarrow (2)$ follows immediately from Proposition 7.3.2.11. Conversely, suppose that there exists a natural transformation $\beta : F_0 \rightarrow F_1 \circ \delta $ which exhibits $F_1$ as a left Kan extension of $F_0$ along $\delta $. By virtue of Remark 7.3.1.9, we can modify $\beta $ by a homotopy and thereby arrange that there exists a functor $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying $F|_{\operatorname{\mathcal{K}}} = F_0$, $F|_{\operatorname{\mathcal{K}}} = F_1$ and for which the induced map
\[ \Delta ^1 \times \operatorname{\mathcal{K}}\simeq \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{K}}} \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}} \]
coincides with $\beta $ (Warning 7.3.2.12). Applying Proposition 7.3.2.11, we see that $F$ is left Kan extended from $\operatorname{\mathcal{K}}$.
$\square$