Remark 7.2.2.4. The converse of Corollary 7.2.2.3 is also true: if $e: A \rightarrow B$ is a morphism of simplicial sets having the property that precomposition with the induced map $e^{\triangleleft }: A^{\triangleleft } \rightarrow B^{\triangleleft }$ carries limit diagrams to limit diagrams, then $e$ is left cofinal. Moreover, it suffices check this condition for diagrams in the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces (see Corollary 7.4.5.14).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$