Theorem 2.6.1 on Weibel's AITAH says:
Let $L:\sf A\to B$ and $R:\sf B\to A$ be an adjoint pair ($L\dashv R$) of additive functors (between abelian categories $\sf A$, $\sf B$). Then $L$ is right-exact and $R$ is left-exact.
My question is: isn't the additivity more a consequence than an hypothesis? I mean, is the theorem still true after the changes made below?
Let $L:\sf A\to B$ and $R:\sf B\to A$ be an adjoint pair ($L\dashv R$) of functors (between abelian categories $\sf A$, $\sf B$). Then $L$, $R$ are additive, $L$ is right-exact and $R$ is left-exact.
I'd say that the second indented sentence is true: being additive is equivalent to preserving finitary direct sums, and since left adjoints preserve colimits and right adjoints preserve limits, any adjoint functor (between abelian categories) preserves finitary direct sums, so it is additive. Hence $L$ is additive and, preserving colimits, $L$ preserves cokernels; the two facts in italics are indeed equivalent to $L$ being right-exact, and I'd argue analogously that $R$ is left-exact.
As I said, it would be strange stating the theorem in the first form, if the additivity was granted, so I'm asking you for a confirmation. Thank you in advance.
You are correct. In fact, this can be generalized even further. For one, your argument works for any adjoint pair between additive categories, not just abelian categories. Furthermore, you can upgrade the conclusion to the adjunction being in fact an additive (or enriched) adjunction, meaning the natural bijections $\mathrm{Hom}(LX,Y)\cong\mathrm{Hom}(X,RY)$ for any objects $X$ of $A$ and $Y$ are $B$ are in fact automatically isomorphisms of abelian groups.