Example 9.2.3.10. Let $X$ be a Kan complex. Then the identity map $\operatorname{id}_{X}: X \rightarrow X$ exhibits $X$ as an $\operatorname{Ind}$-completion of itself. This follows from the criterion of Corollary 9.2.3.6, since every vertex $x \in X$ is compact (see Example 9.2.2.5). More generally, for every pair of regular cardinals $\kappa \leq \lambda $, the identity map $\operatorname{id}_{X}$ exhibits $X$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of itself.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$