Kerodon

Example 7.6.2.31. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. Then, for every object $X \in \operatorname{\mathcal{C}}$, the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ admits finite limits. This follows from Corollary 7.6.2.30, since $\operatorname{\mathcal{C}}_{/X}$ also admits finite pullbacks (Remark 7.6.2.13), and has a final object given by the identity morphism $\operatorname{id}_{X}: X \rightarrow X$ (Proposition 4.6.7.22). Similarly, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor which preserves pullbacks, then the induced functor $F_{/X}: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{D}}_{ /F(X)}$ preserves finite limits.