# Understanding Koszul Duality in BGG and Gelfand, Manin

I'm trying to understand a particular point in the proof of Koszul duality between $D^b(\Lambda(V))$ and $D^b(S(V^*))$ as seen in "Algebraic Bundles over $\mathbb{P}^n$ and Problems of Linear Algebra" by Bernstein, Gelfand and Gelfand. In particular, I am trying to work through Prop. 3, where we introduce functors

$F: C^b(\Lambda(V)) \rightarrow C^b(S(V^*))$ and

$G: C(S(V^*)) \rightarrow C(\Lambda(V))$.

The goal of the proof is to show these functors define an equivalence of derived categories.

I have two particular questions:

1) In the proof, it is first demonstrated that $F$ and $G$ are adjoint. While I can prove this, I do not understand how it helps the argument; I cannot find a source which connects adjoint functors with equivalences on the derived category level.

2) In both BGG and in Gelfand-Manin "Methods of Homological Algebra", the next step is to show that $F$ and $G$ lift to derived functors, and that they form an equivalence of categories; both sources simply say use the Koszul complex for this step. As for the lifting, I know I need to demonstrate that $F$ and $G$ map acyclic complexes to acyclic complexes, but I don't see how to use the acyclicity of the Koszul resolution here. Moreover, once this is done, I'm stuck again on showing that $F_D$ and $G_D$ actually form an equivalence of categories. How does all of this follow from the properties of the Koszul complex?