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Proposition 7.7.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and finite colimits. Then finite colimits in $\operatorname{\mathcal{C}}$ are universal if and only if $\operatorname{\mathcal{C}}$ satisfies the following pair of conditions:
- $(a)$
Pushouts in $\operatorname{\mathcal{C}}$ are universal: that is, $\operatorname{\mathcal{C}}$ satisfies the second Mather cube theorem (Remark 7.7.5.2).
- $(b)$
The initial object $\emptyset \in \operatorname{\mathcal{C}}$ is universal: that is, every morphism $C \rightarrow \emptyset $ is an isomorphism (Example 7.7.1.17).
Proof.
According to Corollary 7.7.2.34, finite colimits in $\operatorname{\mathcal{C}}$ are universal if and only if, for every morphism $f: C' \rightarrow C$ of $\operatorname{\mathcal{C}}$, the pullback functor
\[ f^{\ast }: \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}_{/C'} \quad \quad X \mapsto C' \times _{C} X \]
preserves finite colimits. By virtue of Corollary 7.6.2.30, this is equivalent to the requirement that $f^{\ast }$ preserves both pushouts and initial objects. The desired result follows by allowing $f$ to vary (and using Corollary 7.7.2.34 again).
$\square$