Proposition 7.3.8.19. Let $\operatorname{\mathcal{C}}_0$, $\operatorname{\mathcal{C}}_1$, $\operatorname{\mathcal{C}}_2$, and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Suppose we are given functors $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{D}}$, $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{D}}$, $G: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_1$, and $H: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_2$, where $F_1$ is a left Kan extension of $F_0$ along $G$. The following conditions are equivalent:
- $(1)$
The functor $F_1$ admits a left Kan extension along $H$.
- $(2)$
The functor $F_0$ admits a left Kan extension along $H \circ G$.
Proof.
The implication $(1) \Rightarrow (2)$ is immediate from Proposition 7.3.8.18. To prove the converse, assume that $(2)$ is satisfied. Define $\operatorname{\mathcal{C}}$ as in the proof of Proposition 7.3.8.18. Using the criterion of Corollary 7.3.5.8, we see that $F_0$ admits a left Kan extension $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. It follows from Proposition 7.3.2.11 that $F|_{ \operatorname{\mathcal{C}}_1 }$ is a left Kan extension of $F_0$ along $G$, and is therefore isomorphic to $F_1$ (Remark 7.3.6.6). We may therefore assume without loss of generality that $F_1 = F|_{\operatorname{\mathcal{C}}_1}$ (Remark 7.3.1.10). We will complete the proof by showing that $F_{12} = F|_{\operatorname{\mathcal{C}}_{12}}$ is left Kan extended from $\operatorname{\mathcal{C}}_{1}$, and therefore exhibits $F|_{\operatorname{\mathcal{C}}_2}$ as a left Kan extension of $F_1$ along $H$ (Proposition 7.3.2.11).
Fix an object $X \in \operatorname{\mathcal{C}}_2$, and let $F_{X}$ denote the composite map
\[ (\operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X})^{\triangleright } \hookrightarrow ( \operatorname{\mathcal{C}}_{/X} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}. \]
We wish to show that the composite map
\[ (\operatorname{\mathcal{C}}_{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X})^{\triangleright } \xrightarrow {F_ X} \operatorname{\mathcal{D}} \]
is a colimit diagram in $\operatorname{\mathcal{D}}$. As in the proof of Proposition 7.3.8.18, the inclusion map $\operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} \hookrightarrow \operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X}$ is right cofinal. It will therefore suffice to show that $F_{X}$ is a colimit diagram in $\operatorname{\mathcal{D}}$ (Corollary 7.2.2.3). This is clear: by construction, the functor $F$ is left Kan extended from the full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$, and is therefore also left Kan extended from the larger subcategory $\operatorname{\mathcal{C}}_{01} \subseteq \operatorname{\mathcal{C}}$ (Proposition 7.3.8.6).
$\square$