Lemma 4.18.6. Let $\mathcal{C}$ be a category. The following are equivalent:

Nonempty finite colimits exist in $\mathcal{C}$.

Coproducts of pairs and coequalizers exist in $\mathcal{C}$.

Coproducts of pairs and pushouts exist in $\mathcal{C}$.

Lemma 4.18.6. Let $\mathcal{C}$ be a category. The following are equivalent:

Nonempty finite colimits exist in $\mathcal{C}$.

Coproducts of pairs and coequalizers exist in $\mathcal{C}$.

Coproducts of pairs and pushouts exist in $\mathcal{C}$.

**Proof.**
Omitted. Hint: This is the dual of Lemma 4.18.3.
$\square$

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