Kerodon

Warning 7.1.4.6. In the formulation of Definition 7.1.4.4, 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 7.1.4.4 in cases where the $\infty $-category admits $K$-indexed limits or colimits, so that the conclusion of Definition 7.1.4.4 is non-vacuous.