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

Warning In the formulation of Definition, it is not necessary to assume that the $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits or colimits. For example, if $\operatorname{\mathcal{C}}$ is an $\infty $-category which contains no limit diagrams $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed limits. In practice, we will usually (but not always) apply the terminology of Definition in cases where the $\infty $-category admits $K$-indexed limits or colimits, so that the conclusion of Definition is non-vacuous.