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Remark 7.1.4.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $g: B \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and suppose we are given a diagram $\overline{f}: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{/g}$, which we can identify with a morphism of simplicial sets

\[ \overline{q}: (A \star B)^{\triangleleft } \simeq A^{\triangleleft } \star B \rightarrow \operatorname{\mathcal{C}}. \]

Then $\overline{f}$ is a limit diagram in the slice $\infty $-category $\operatorname{\mathcal{C}}_{/g}$ if and only if $\overline{q}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.