Proposition 7.1.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $f: K \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the composite diagram
\[ K \xrightarrow {f} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]
admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}$. Then:
- $(1)$
The diagram $f$ admits a colimit in $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$.
- $(2)$
Let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be an extension of $f$. Then $\overline{f}$ is a colimit diagram if and only if, for every vertex $b \in B$, the morphism
\[ K^{\triangleright } \xrightarrow { \overline{f} } \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]
is a colimit diagram in $\operatorname{\mathcal{C}}$.
Proof of Proposition 7.1.6.1.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $f: K \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the composite diagram
\[ K \xrightarrow {f} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]
admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}$. Applying Corollary 7.1.6.8, we see that $f$ admits an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ with the property that, for every vertex $b \in B$, the composition $\operatorname{ev}_{b} \circ \overline{f}$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Applying Corollary 7.1.6.12, we see any such extension is a colimit diagram in $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$. To complete the proof, it will suffice to show the converse: if $\overline{f}': K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ is any colimit diagram extending $f$ and $b \in B$ is a vertex, then $\operatorname{ev}_{b} \circ \overline{f}'$ is also a colimit diagram in $\operatorname{\mathcal{C}}$. In this case, the extension $\overline{f}'$ is isomorphic to $\overline{f}$ as an object of the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{Fun}(B,\operatorname{\mathcal{C}}) )$. It follows that $\operatorname{ev}_{b} \circ \overline{f}'$ is isomorphic to $\operatorname{ev}_{b} \circ \overline{f}$ as an object of the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ and therefore a colimit diagram by virtue of Corollary 7.1.2.14.
$\square$