First a remark, I skipped the hypothesis "left adjoint to an exact functor" on purpose because the sketch of argument I wrote down I didn't use this, at least according to me. I know that there should be an error somewhere but I can't find out where
Let $F$ be our left adjoint functor and let $G$ it's right adjoint (defined between two abelian cats). Let $A$ be a projective object in the respective category. By hypothesis We have a natural isomorphism $$ \hom(F(A),B) \cong \hom(A,G(B)) $$
Being $A$ projective, the second Hom functor is exact and therefore the first is exact too and therefore $F(A)$ is projective. Now the only doubt is that the natural isomorphism between the two doesn't preserve exactness, but this is strange because exactness is a "functorial property".