Let $T$ and $S$ be triangulated categories. Let $F:T\rightarrow S$ and $G:S\rightarrow T$ be two adjoint functors. Assume that one of them is exact (i.e. sends exact triangles to exact triangles and commutes with the suspension functor), then why is the other also exact?
I think I have a proof using Yoneda's lemma that if one commutes with suspensions then so does the other, but I don't know how to prove that if one sends exact triangles to exact triangles then so does the other, any help?