Warning 7.1.7.16. The converse of Corollary 7.1.7.15 is false in general. If the $\infty $-category $\operatorname{\mathcal{C}}$ is not assumed to admit $K$-indexed colimits, then it is possible to have a colimit diagram $K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ which is not a levelwise colimit diagram.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$