Kerodon

Variant 7.7.5.7. Let $\kappa $ be an infinite cardinal, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and $\kappa $-small colimits. Then $\kappa $-small colimits in $\operatorname{\mathcal{C}}$ are universal if and only if $\kappa $-small coproducts in $\operatorname{\mathcal{C}}$ are universal and $\operatorname{\mathcal{C}}$ satisfies the second Mather cube theorem. This follows from the proof of Proposition 7.7.5.6, using Exercise 7.6.6.11 in place of Corollary 7.6.2.30.