Prove that an (additive) functor $F$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives

Prove that an (additive) functor $$F$$ between abelian categories (categories of modules) that admits an exact left adjoint must preserve injectives. State and prove the dual result.

I have no idea on how to solve this question. I know that right adjoints preserve limits and left adjoints preserve colimits, so it suffices to show that every injective module is a limit. Why is this true? What would also be the dual statement? that every additive functor $$G$$ between abelian categories that admits an exact right adjoint must preserve projectives?

Let $$F:\mathcal{D}\to \mathcal{C}$$ and let $$G$$ its left adjoint.
Take $$I\in \mathcal{D}$$ injective. You want to see that $$F(I)\in \mathcal{C}$$ is injective.
The adjunction gives you an isomorphism $$\mathcal{C}(-,F(I))\cong\mathcal{D}(G(-),I)$$ which proves that the functor $$\mathcal{C}(-,F(I))$$ is exact as it is (isomorphic to) the composition of exact functors ($$G$$ by assumption and $$\mathcal{D}(-,I)$$ since $$I$$ is injective).
• Thank you, Why is composition of exact functors exact? Edit: Oh okay I see, it's because if $L \to M \to N$ is exact then $F_1(L) \to F_1(M) \to F_1(N)$ is exact and so $F_2 \circ F_1(L) \to F_2 \circ F_1(M) \to F_2 \circ F_1(N)$ is exact, if $F_1, F_2$ are exact functors. Thanks! Commented Mar 20 at 5:15