Remark 9.3.2.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category.
If $\operatorname{\mathcal{C}}$ has a final object, then the partially ordered set $\operatorname{Sub}(\operatorname{\mathcal{C}})$ has a largest element: namely, the isomorphism class $[X]$, where $X$ is any final object of $\operatorname{\mathcal{C}}$.
If $\operatorname{\mathcal{C}}$ admits finite products, then $\operatorname{Sub}(\operatorname{\mathcal{C}})$ is a lower semilattice: that is, every finite subset of $\operatorname{Sub}(\operatorname{\mathcal{C}})$ has a greatest lower bound. In particular, every pair of elements $[X], [Y] \in \operatorname{Sub}(\operatorname{\mathcal{C}})$ have a greatest lower bound which we will denote by $[X] \cap [Y]$, given by the isomorphism class of the product $X \times Y$.