Lemma 4.18.7 (002Q)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.18: Finite limits and colimits
Lemma 4.18.7 (cite)

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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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