Corollary 7.2.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Then a diagram $f: B \rightarrow \operatorname{\mathcal{C}}$ has a limit if and only if the composite diagram $(f \circ e): A \rightarrow \operatorname{\mathcal{C}}$ has a limit.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. If $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram extending $f$, then Corollary 7.2.2.3 guarantees that $\overline{f} \circ e^{\triangleleft }: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram extending $f \circ e$. Conversely, if $f \circ e$ can be extended to a colimit diagram, then Proposition 7.2.2.9 (applied in the special case $\operatorname{\mathcal{D}}= \Delta ^0$) guarantees that $f$ can also be extended to a colimit diagram. $\square$