$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.7.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and finite colimits. Then finite 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 the first and second Mather cube theorems.
- $(b)$
The initial object $\emptyset \in \operatorname{\mathcal{C}}$ is universal: that is, every morphism $C \rightarrow \emptyset $ is an isomorphism.
Proof.
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 finite colimits in $\operatorname{\mathcal{C}}$ are strongly universal if and only if the functor $\operatorname{Tr}$ preserves finite limits (Remark 7.7.2.18). By virtue of Corollary 7.6.2.30, this is equivalent to the requirement that $\operatorname{Tr}$ preserves pullback squares and final objects. The case of pullback squares is a reformulation of condition $(a)$ (Remark 7.7.5.2), and the case of final objects is a reformulation of condition $(b)$ (Example 7.7.2.16).
$\square$