$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.7.5.9. 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 strongly universal if and only if the following conditions are satisfied:
- $(a)$
The $\infty $-category $\operatorname{\mathcal{C}}$ satisfies both Mather cube theorems.
- $(b)$
In the $\infty $-category $\operatorname{\mathcal{C}}$, $\kappa $-small coproducts are universal.
Proof.
By virtue of Proposition 7.7.5.8, we may assume without loss of generality that finite colimits in $\operatorname{\mathcal{C}}$ are universal, so that $\operatorname{\mathcal{C}}$ satisfies both Mather cube theorems. Choose an uncountable cardinal $\lambda $ for which $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, and let
\[ \operatorname{Tr}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}^{< \lambda } \quad \quad C \mapsto \operatorname{\mathcal{C}}_{/C} \]
be a contravariant transport representation for the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$. Then $\kappa $-small colimits in $\operatorname{\mathcal{C}}$ are strongly universal if and only if the functor $\operatorname{Tr}$ preserves $\kappa $-small limits (Remark 7.7.2.18). By virtue of Exercise 7.6.6.11, this is equivalent to the requirement that $\operatorname{Tr}$ preserves $\kappa $-small products: that is, that $\kappa $-small coproducts in $\operatorname{\mathcal{C}}$ are strongly universal. Using Corollary 7.7.4.11, we see that this is equivalent to condition $(b)$.
$\square$