Variant 7.1.1.16. It will often be useful to extend the terminology of Definition 7.1.1.14, replacing the individual simplicial set $K$ by a collection of simplicial sets. For example:
We say that an $\infty $-category $\operatorname{\mathcal{C}}$ admits finite limits if it admits $K$-indexed limits for every finite simplicial set $K$ (Definition 3.6.1.1), every diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ admits a limit.
We say that an $\infty $-category $\operatorname{\mathcal{C}}$ admits finite colimits if it admits $K$-indexed colimits for every finiite simplicial set $K$.
We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is complete if it admits $K$-indexed limits for every small simplicial set $K$.
We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete if it admits $K$-indexed colimits for every small simplicial set $K$.