A contravariant functor $F : C \to C'$ between categories $C$ and $C'$ is a dual equivalence of categories if there exists another functor $G : C' \to C$ such that $F \circ G \cong 1_{C'}$ and $G \circ F \cong 1_{C}$, where $1_{C}$ and $1_{C'}$ are the identity functors. (The more precise definition is in terms of natural transformations.)
A functor $F$ is called exact if it preserves short exact sequences (but reverses the order of the morphisms in the sequence). I've read in numerous places that any dual equivalences of categories is an exact functor. However I can't see why this is necessarily the case, and can't seem to find a reference.