$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.3.8.6 (Transitivity for Kan Extensions). Let $\overline{F}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}\subseteq \overline{\operatorname{\mathcal{C}}}$ be full subcategories. Then $\overline{F}$ is $U$-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 $U$-left Kan extended from $\operatorname{\mathcal{C}}$.
- $(2)$
The restriction $\overline{F}|_{\operatorname{\mathcal{C}}}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.
Proof of Proposition 7.3.8.6.
It follows immediately from the definitions that if $\overline{F}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$, then the functor $F = \overline{F}|_{\operatorname{\mathcal{C}}}$ has the same property. We may therefore assume that condition $(2)$ is satisfied. Fix an object $X \in \overline{\operatorname{\mathcal{C}}}$. We will complete the proof by showing that $\overline{F}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ at $X$ if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}$ at $X$. Let $\overline{F}_{X}$ denote the composite map
\[ (\operatorname{\mathcal{C}}\times _{\overline{\operatorname{\mathcal{C}}}} \overline{\operatorname{\mathcal{C}}}_{/X})^{\triangleright } (\overline{\operatorname{\mathcal{C}}}_{/X})^{\triangleright } \rightarrow \overline{\operatorname{\mathcal{C}}} \xrightarrow { \overline{F} } \operatorname{\mathcal{D}}. \]
We wish to show that $\overline{F}_{X}$ is a $U$-colimit diagram if and only if its restriction to $(\operatorname{\mathcal{C}}^{0} \times _{\overline{\operatorname{\mathcal{C}}}} \overline{\operatorname{\mathcal{C}}}_{/X})^{\triangleright }$ is a $U$-colimit diagram. Let $F_{X}$ denote the restriction of $\overline{F}_{X}$ to $\operatorname{\mathcal{C}}\times _{\overline{\operatorname{\mathcal{C}}}} \overline{\operatorname{\mathcal{C}}}_{/X}$. By virtue of Proposition 7.3.8.1, it will suffice to show that $F_{X}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0} \times _{\overline{\operatorname{\mathcal{C}}}} \overline{\operatorname{\mathcal{C}}}_{/X}$. This follows by applying Corollary 7.3.8.4 to the right fibration $\operatorname{\mathcal{C}}\times _{ \overline{\operatorname{\mathcal{C}}} } \overline{\operatorname{\mathcal{C}}}_{/X} \rightarrow \operatorname{\mathcal{C}}$.
$\square$