The Stacks project

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

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

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

  3. 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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