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

Question

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?

Answer 1

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).

You are right about the dual statement.