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This version of Koszul duality (as well as many others) can be deduced from the Barr-Beck-Lurie theorem (cf. Lurie's book Higher Algebra). You can consider the functor from k-mod to A-mod of tensoring by V and ask for it to have a left or right adjoint, giving rise to a comonad or monad on k-mod (ie the coalgebra or algebra form). These are two finiteness conditions on V (k-dualizability and A-dualizability - ie dualizability as a vector space or compactness as A-module - though maybe I'm mixing up the order). If you ask for both simultaneously then you guarantee that the functor from A-mod to (co)modules over the (co)monad is both limit and colimit preserving -- this is overkill but an easy way to ensure the hypotheses of the Barr-Beck-Lurie theorem are satisfied. You now get an equivalence between your category of (co)modules and a COMPLETION of A-mod --- i.e., the part of A-mod that your particular chosen module V sees (the category it generates). (This is how you satisfy the other requirement of the theorem, i.e., that the corresponding functor is conservative).

The classical version of this is A=S, symmetric algebra in one variable, ie A-mod = quasicoherent sheaves on the line, and V is the augmentation module (skyscraper at the origin). We then have a descent theorem, saying that the completion of A at the augmentation is derived equivalent to sheaves on a point with descent data for inclusion of point to line -- which is exactly modules for the Koszul dual exterior coalgebra (or dually exterior algebra).

EDIT: In response to the comment about "seeing" modules: technically the condition is whether there are any Exts between your given object V and some other object W. I think of this geometrically: if V stands for a skyscraper on a variety, it will see the entire formal neighborhood of the point, i.e. intuitively you can manufacture Exts with the skyscraper for anything that has a stalk at this point. So for instance for enveloping algebras a representation will see only representations that have all invariants the same as V (i.e. the center must act with the same generalized character). So I'm not sure I understand the question regarding Lie algebras (take the example of $\mathfrak g$ the trivial one-dimensional Lie algebra and we're back in the original Koszul duality discussed in the paragraph above).

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This version of Koszul duality (as well as many others) can be deduced from the Barr-Beck-Lurie theorem (cf. Lurie's book Higher Algebra). You can consider the functor from k-mod to A-mod of tensoring by V and ask for it to have a left or right adjoint, giving rise to a comonad or monad on k-mod (ie the coalgebra or algebra form). These are two finiteness conditions on V (k-dualizability and A-dualizability - ie dualizability as a vector space or compactness as A-module - though maybe I'm mixing up the order). If you ask for both simultaneously then you guarantee that the functor from A-mod to (co)modules over the (co)monad is both limit and colimit preserving -- this is overkill but an easy way to ensure the hypotheses of the Barr-Beck-Lurie theorem are satisfied. You now get an equivalence between your category of (co)modules and a COMPLETION of A-mod --- i.e., the part of A-mod that your particular chosen module V sees (the category it generates). (This is how you satisfy the other requirement of the theorem, i.e., that the corresponding functor is conservative).

The classical version of this is A=S, symmetric algebra in one variable, ie A-mod = quasicoherent sheaves on the line, and V is the augmentation module (skyscraper at the origin). We then have a descent theorem, saying that the completion of A at the augmentation is derived equivalent to sheaves on a point with descent data for inclusion of point to line -- which is exactly modules for the Koszul dual exterior coalgebra (or dually exterior algebra).