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 $H = \lbrace H^n, n \in \mathbb{N}$ 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 given any delta functor $K = \lbrace K^n, n \in \mathbb{N} \rbrace$ (not necessarily with $K^0 = \Gamma$ !) and a morphism $f^0:H^0 \rightarrow K^0$ in degree zero, then there exists a unique extension of $f^0$ to a morphism $f:H \rightarrow K$ 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).
The universality condition forces a strong connection between $\Gamma$ and $H$. This allows you for example to extend constructions in degree 0 to arbitrary degree. An example from group cohomology: Let U be a subgroup of a group G and let $g \in G$. Then conjugation is an isomorphism of functors $(-)^U \rightarrow (-)^{gUg^{-1}}$, where $(-)^U$ is the U-invariant functor on G-modules. Now, $H^n(U,-)$ is a universal cohomology theory for $(-)^U$ and $H^n(gUg^{-1},-)$ is a universal cohomology theory for $(-)^{gUg^{-1}}$. Hence, the exists a unique isomorphism of delta-functors $H^n(U,-) \rightarrow H^n(gUg^{-1},-)$ and in this way we have extended the conjugation to all cohomology groups. This is very inexplicit of course, but this is a powerful property for more advanced constructions. And verifying each time that constructions are compatible with the deltas is not a nice job (normally people simply say, it works).
In Lang's book "Topics in the Cohomology of Groups" you will find a lot more of such arguments.
If you now want to get rid of all choices, you can move to the derived category.
Hope this helps.