Question on adjoint functors Can someone provide me an enlightenment on the following three statements?
(I stumbled on them at the part dealing injective modules in a text of homological algebra.)

1) Let $F \dashv G \colon \mathcal{C} \to \mathcal{D}$ is a pair of adjoint functors and $F$ preserves monomorphisms. Then $G$ preserves injectives.
2) A left adjoint preserves epimorphisms.
3) A right adjoint preserves monomorphisms.

Seeing that the text gives no clues for the proofs, they must make an easy exercise.
But I cannot make out how to show them.  
 A: General advice when dealing with this sort of exercise: Definitions, definitions, definitions. Always go back to the definitions. Write down explicitly what you want to prove, and in 95% of cases the proof will almost write itself down for you.

For example for question #1: I will write down everything very formally. In a normal proof everything would be condensed in a few lines (cf. Hanno's answer), but hopefully this will help you in finding how to reason.
What is an injective? It's an object $I$ such that if for every morphism $f : A \to I$ and every monomorphism (notice how $F$ conveniently preserves monomorphisms?) $g : A \to B$, there exists a morphism $h : B \to I$ such that $h \circ g = f$ (write down the commutative diagram).
Now you want to show that if 


*

*Data 1: $I \in \mathcal{D}$ is injective,


then $G(I) \in \mathcal{C}$ is injective. So by definition you take 


*

*Data 2: any morphism $f : A \to G(I)$, and

*Data 3: any monomorphism $g : A \to B$,


and you look for some $h : B \to G(I)$ that makes the diagram commute.
Now you only have two hypotheses:


*

*Hyp 4: $F \dashv G$,

*Hyp 5: $F$ preserves monomorphisms.


There's only one monomorphism in this story, so apply your Hyp 5 to it:


*

*Fact 6: $F(g) : F(A) \to F(B)$ is a monomorphism.


You know that $I$ is injective. By Hyp 4, you've got a morphism $\tilde{f} : F(A) \to I$ naturally associated to $f$. By taking Fact 6 into account, and the Data 1 that $I$ is injective, you find that


*

*Fact 7: there exists a morphism $\phi : F(B) \to I$ such that $\phi \circ F(g) = \tilde{f}$.


Now you apply Hyp 4 again and you find a morphism $h = \bar\phi : B \to G(I)$. Since the adjunction $F \dashv G$ is natural, you finally find that $h \circ g = f$. Qed.

Now if you follow the same pattern of reasoning you should find the answers to question 2 and 3.
A: Hint for (1): Note that an object $I\in{\mathscr D}$ is injective if and only if ${\mathscr D}(-,I): {\mathscr D}^{\text{op}}\to\textsf{Set}$ preserves epimorphisms (i.e., turns monomorphisms in ${\mathscr D}$ into surjective maps of sets) - and analogously for objects of ${\mathscr C}$. Now suppose $X\in{\mathscr D}$ is injective & that ${\mathscr F}$ preserves monomorphisms and check that by adjointness of ${\mathscr F}$ and ${\mathscr G}$ the functor ${\mathscr C}(-,{\mathscr G}X)$ also maps monomorphisms in ${\mathscr C}$ to epimorphisms.
Parts (2) and (3) are similar, noting that a morphism $f: X\to Y$ in ${\mathscr C}$ is, say, an epimorphism if and only if for all $Z\in{\mathscr C}$ the induced map of sets ${\mathscr C}(f,Z): {\mathscr C}(Y,Z)\to{\mathscr C}(X,Z)$ is injective, i.e. a monomorphism in $\textsf{Set}$. Similar for monomorphisms.
