$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.3.8.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ be diagrams, and let $\alpha : F_0 \rightarrow F \circ \delta $ be a natural transformation which exhibits $F$ as a left Kan extension of $F_0$ along $\delta $ (see Variant 7.3.1.5). Then:
- $(1)$
The diagram $F$ admits a colimit in $\operatorname{\mathcal{D}}$ if and only if $F_0$ admits a colimit in $\operatorname{\mathcal{D}}$.
- $(2)$
Let $X$ be an object of $\operatorname{\mathcal{D}}$, let $\underline{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ denote the constant functor taking the value $X$. Then a natural transformation $\beta : F \rightarrow \underline{X}$ exhibits $X$ as a colimit of the diagram $F$ if and only if the composite natural transformation
\[ F_0 \xrightarrow {\alpha } F \circ \delta \xrightarrow { \beta |_{K} } \underline{X}|_{ K } \]
exhibits $X$ as a colimit of the diagram $F_0$.
Proof.
Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $i: K \hookrightarrow \operatorname{\mathcal{K}}$, where $\operatorname{\mathcal{K}}$ is an $\infty $-category. Since $\operatorname{\mathcal{C}}$ is an $\infty $-category, we extend $\delta $ and $F_0$ to functors $\overline{\delta }: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ and $\overline{F}_0: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{D}}$, respectively. Similarly, we can extend $\alpha $ to a natural transformation $\overline{\alpha }: \overline{F}_0 \rightarrow F \circ \overline{\delta }$. It follows from Proposition 7.3.1.15 that we $\overline{\alpha }$ exhibits $F$ as a left Kan extension of $\overline{F}_0$ along $\overline{\delta }$. We may therefore replace $K$ by $\operatorname{\mathcal{K}}$ and thereby reduce to proving Corollary 7.3.8.20 in the special case where $K$ is an $\infty $-category. In this case, assertion $(1)$ is a special case of Proposition 7.3.8.19, and assertion $(2)$ is a special case of Proposition 7.3.8.18 (see Example 7.3.1.7).
$\square$