I am trying to find a proof of the following statement:
Let $A$ and $B$ be abelian categories, and let $(F, G)$ be an adjoint pair of additive functors $F : A → B$ and $G: B → A$. Show that $F$ is right exact and $G$ is left exact.
I understand the proof when $A$ and $B$ are module categories, but the proof I have seen for this case uses left exxactness of the Hom functors together with the isomorphism $\text{Hom}_{R}(R, M) \cong M$ for all $R$-modules $M$. I don't think this can be done in the general case.
I can't seem to find a anything in the general case. I thought of using the unit and counit in some way, but I literally have no clue how to prove this.