The Stacks project

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:

  1. $F$ is right exact,

  2. $F$ commutes with finite coproducts and coequalizers, and

  3. $F$ 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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