Definition 7.6.6.6. Let $\kappa $ be an infinite cardinal. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-complete if admits $K$-indexed limits, for every $\kappa $-small simplicial set $K$. We say that $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete if it admits $K$-indexed colimits, for every $\kappa $-small simplicial set $K$.
We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-continuous if it preserves $K$-indexed limits, for every $\kappa $-small simplicial set $K$. We say that $F$ is $\kappa $-cocontinuous if it preserves $K$-indexed colimits, for every $\kappa $-small simplicial set $K$.