Variant 7.6.6.7. Let $\kappa $ be an infinite cardinal. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete if it admits $K$-indexed colimits, for every $\kappa $-small simplicial set $K$. Equivalently, the $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete if the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is $\kappa $-complete.
We say that a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\kappa $-small colimits if it preserves $K$-indexed colimits, for every $\kappa $-small simplicial set $K$. Equivalently, $F$ preserves $\kappa $-small colimits if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ preserves $\kappa $-small limits.