# Kerodon

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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.