Variant 7.1.4.9. It will often be useful to extend the terminology of Definition 7.1.4.4, replacing the individual simplicial set $K$ by a collection of simplicial sets.
We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves finite limits if it preserves $K$-indexed limits, for every finite simplicial set $K$.
We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves finite colimits if it preserves $K$-indexed colimits, for every finite simplicial set $K$.
We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves small limits if it preserves $K$-indexed limits, for every small simplicial set $K$.
We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves small colimits if it preserves $K$-indexed colimits, for every small simplicial set $K$.