Lemma 4.23.3 (0GMN)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.23: Exact functors
Lemma 4.23.3 (cite)

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Lemma 4.23.3. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. Suppose all finite colimits exist in $\mathcal{A}$ , see Lemma 4.18.7. The following are equivalent:

$F$ is right exact,
$F$ commutes with finite coproducts and coequalizers, and
$F$ transforms an initial object of $\mathcal{A}$ into an initial object of $\mathcal{B}$ , and commutes with pushouts.

Proof. Dual to Lemma 4.23.2. $\square$

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