Properties of Adjoint functors

Suppose that $(F,G)$ is an adjoint pair of covariant functors. I read in a book the folowing statement

"if the right adjoint preserves epimorphisms, then the left adjoint preserves projectives."

Can anyone give me a hint as how to prove this statement?

I need the result, but I don't want to use it without proving it first!

Thanks!

As with all other such general statements, there is only one possible proof and you can write it down just by unwinding the definitions.

Let $F : C \to D$ be left adjoint to $G : D \to C$. Assume that $G$ preserves epimorphisms. Let $x \in C$ be projective. We want to show that $F(x) \in D$ is projective. This means that for all epimorphisms $y \to y'$ in $D$ the map $\hom(F(x),y) \to \hom(F(x),y')$ is surjective. This map identifies with $\hom(x,G(y)) \to \hom(x,G(y'))$, which is induced by $G(y) \to G(y')$. By assumption, this morphism is an epimorphism. Since $x$ is projective, the claim follows.

By the way, the dual statement is also useful: If $F$ is left adjoint to $G$, and $F$ preserves monomorphisms, then $G$ preserves injective objects. This is useful, for example, in the context of sheaf cohomology, where one needs that the restriction of an injective sheaf to an open subset is still injective.