Lemma 4.18.7. Let $\mathcal{C}$ be a category. The following are equivalent:

Finite colimits exist in $\mathcal{C}$.

Finite coproducts and coequalizers exist in $\mathcal{C}$.

The category has an initial object and pushouts exist.

Lemma 4.18.7. Let $\mathcal{C}$ be a category. The following are equivalent:

Finite colimits exist in $\mathcal{C}$.

Finite coproducts and coequalizers exist in $\mathcal{C}$.

The category has an initial object and pushouts exist.

**Proof.**
Omitted. Hint: This is dual to Lemma 4.18.4.
$\square$

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