Remark 7.6.6.17. In the situation of Proposition 7.6.6.16, suppose that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories which preserves the product of the collection $\{ X(n) \} _{n \geq 0}$. Then $F$ preserves limits of the tower $X$ if and only if it preserves equalizers of the pair of morphisms $f, \operatorname{id}_{P}: P \rightarrow P$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$