In my opinion, all answers go a little too far. In this (non-topological!) setting I think about this as follows: The aim is to analyze the lack of right-exactness of a left-exact functor $\Gamma:C \rightarrow D$ between abelian categories. One functorial notion that captures this inexactness is a delta-functor $\lbrace H^n, n \in \mathbb{N}$ with $H^0 = \Gamma$. This is what I call a cohomology theory for $\Gamma$. Now, there may exist a lot of them, and so it makes sense to look for universal cohomology theories for $\Gamma$, where universal means, that any morphism in degree 0 already uniquely extends to a morphism of delta-functors. Up to canonical isomorphism there exists only one universal cohomology theory for $\Gamma$ which is then called the right derivative of $\Gamma$. The question is of course if such a universal cohomology theory for $\Gamma$ exists. And here it comes: If the category $C$ has enough injectives, then $\Gamma$ has a right derivative which can be computed by injective resolutions. (You can confer Lang's Algebra book for all the above notions, it's actually pretty nice).
If you now want to get rid of all choices, you can move to the derived category.
Hope this helps.