Remark 9.3.4.30. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits fiber products. Then, for every object $X \in \operatorname{\mathcal{C}}$, the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ admits finite products (Corollary 7.6.2.17). It follows that the partially ordered set $\operatorname{Sub}(X)$ is a lower semilattice (see Remark 9.3.2.21). In particular, every pair of objects $[X_0], [X_1] \in \operatorname{Sub}(X)$ have a greatest lower bound $[X_0] \cap [X_1]$ in $\operatorname{Sub}(X)$, given concretely by the isomorphism class of the fiber product $[X_0 \times _{X} X_1]$.
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