Variant 7.6.7.6. Let $\kappa $ be an infinite cardinal. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ admits $\kappa $-small colimits if it admits $K$-indexed colimits, for every $\kappa $-small simplicial set $K$. Equivalently, the $\infty $-category $\operatorname{\mathcal{C}}$ admits $\kappa $-small colimits if the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ admits $\kappa $-small limits.
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.