$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Definition Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Suppose we are given a collection $S$ of objects of $\operatorname{\mathcal{C}}$ and a collection $T$ of morphisms of $\operatorname{\mathcal{C}}$ satisfying the assumptions of Corollary, so that there exists a unique subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ whose objects are the elements of $S$ and whose morphisms are the elements of $T$. In this case, we will refer to $\operatorname{\mathcal{C}}'$ as the subcategory of $\operatorname{\mathcal{C}}$ spanned by the objects of $S$ and the morphisms of $T$.