Proposition 9.2.1.1. An $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete if and only if it satisfies both of the following conditions:
The $\infty $-category $\operatorname{\mathcal{C}}$ is finitely cocomplete: that is, it admits finite colimits.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits small filtered colimits.
Proof. Assume that conditions $(a)$ and $(b)$ are satisfied; we will show that $\operatorname{\mathcal{C}}$ is cocomplete (the converse is immediate from the definitions). Let $K$ be a small simplicial set; we wish to show that $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits. Let $\{ K_{\alpha } \} _{\alpha \in A}$ be the collection of all finite simplicial subsets of $K$. For $\alpha ,\beta \in A$, let us write $\alpha \leq \beta $ if $K_{\alpha }$ is contained in $K_{\beta }$. Then $(A, \leq )$ is a small directed partially ordered set, so $\operatorname{\mathcal{C}}$ admits $\operatorname{N}_{\bullet }(A)$-indexed colimits by virtue of condition $(b)$. It follows from condition $(a)$ that $\operatorname{\mathcal{C}}$ also admits $K_{\alpha }$-indexed colimits, for each $\alpha \in A$. Applying Corollary 9.1.6.6, we conclude that $\operatorname{\mathcal{C}}$ admits colimits indexed by $K = \varinjlim _{\alpha \in A} K_{\alpha }$, as desired. $\square$