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Derrived Derived functors vs universal delta functors

I would like to understand the relationship between the derived category definition of a right derived functor Rf (which involves an initial natural transformation n: Qf → (Rf)Q, where Q is the map to the derived category) and the "universal delta functor" definition given in Hartshorne III.1.

I already know that R^if(A) = H^i(Rf(A)). What I want to know most is:

What is the role ole of the natural transformation n in this comparison?

I guess it can be thought of as a natural map from a injective resolution of f(A) to f(an injective resolution of A), but I'm not sure what is the significance of this... Does anyone know a good reference explaining such things?
show/hide this revision's text 1

Derrived functors vs universal delta functors

I would like to understand the relationship between the derived category definition of a right derived functor Rf (which involves an initial natural transformation n: Qf → (Rf)Q, where Q is the map to the derived category) and the "universal delta functor" definition given in Hartshorne III.1.

I already know that R^if(A) = H^i(Rf(A)). What I want to know most is:

What is the role ole of the natural transformation n in this comparison?

I guess it can be thought of as a natural map from a injective resolution of f(A) to f(an injective resolution of A), but I'm not sure what is the significance of this... Does anyone know a good reference explaining such things?