Corollary 7.2.2.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets, and let $f: B \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then an object $X \in \operatorname{\mathcal{C}}$ is a limit of $f$ if and only if it is a limit of the diagram $(f \circ e): A \rightarrow \operatorname{\mathcal{C}}$.
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Proof. If an object $X \in \operatorname{\mathcal{C}}$ is a limit of $f$, then we can choose a limit diagram $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ carrying the cone point of $f^{\triangleleft }$ to the object $X$. Applying Corollary 7.2.2.10, we deduce that $\overline{f} \circ e^{\triangleleft }$ exhibits $X$ as a limit of the diagram $f \circ e$. Conversely, if $X$ is a limit of the diagram $f \circ e$, then Corollary 7.2.2.10 guarantees that the diagram $f$ admits a limit $Y \in \operatorname{\mathcal{C}}$. The preceding argument shows that $Y$ is also a limit of the diagram $f \circ e$. Applying Proposition 7.1.1.12, we deduce that $Y$ is isomorphic to $X$, so that $X$ is also a limit of the diagram $f$. $\square$