Warning 7.7.2.9. Beware that the second assertion of Theorem 7.7.2.8 is generally false if we do not assume either that $\operatorname{\mathcal{C}}$ has $K$-indexed colimits or a final object. For example, suppose that $\operatorname{\mathcal{C}}$ is a Kan complex. Then the evaluation map $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration (Corollary 3.1.3.6), so every morphism $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is an (effective) descent diagram for $\operatorname{ev}_{1}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$