Corollary 7.2.2.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then a morphism of simplicial sets $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram if and only if the composite map
\[ A^{\triangleleft } \xrightarrow {e^{\triangleleft }} B^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]
is a limit diagram.