Remark 9.3.2.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be a discrete object of $\operatorname{\mathcal{C}}$, and suppose we are given a diagram $F: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{/X}$. If the simplicial set $K$ is connected, then $F$ is a limit diagram if and only if its image in $\operatorname{\mathcal{C}}$ is a limit diagram. In particular, the forgetful functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed colimits. If the object $X$ is subterminal, then these conclusions hold more generally if when the simplicial set. See Corollary 9.3.1.22.
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