Occasionally in math I come across constructions or tools that are a bit convoluted. I can look at these constructions and see that they indeed perform the task they were made to do, but sometimes I can't see why they $\textit{should}$ perform this task; why is that the logical thing to choose for the job? Right now, I'm having this issue with the satellite of a functor.

Just to recall, given an additive functor between two abelian categories $F:\mathcal{C} \rightarrow \mathcal{D}$, the satellite is another functor $S_-^1(F): \mathcal{C} \rightarrow \mathcal{D}$ defined by

$S_-^1(F)(M) = lim(ker(F(M) \rightarrow F(P)))$

where $0 \rightarrow M \rightarrow P \rightarrow N \rightarrow 0$ is an exact sequence with $P$ a projective object. Then a derived functor is formed by taking iterations of the satellite: $S_-^n(F) = S_-(S_-^{n-1}(F))$. More information can be found on nlab.

I am learning about derived functors in a slightly different setting, namely with nonadditive categories where there are not necessarily enough projectives in the category $\mathcal{C}$, and so the definition is modified slightly; perhaps the definition is more transparent in the standard setting.

So my question is

Is there any intuition for why the satellite is the correct tool to use for obtaining derived functors? If I needed to create a derived functor out of a given functor, is there a logical progression that would lead me to define the satellite?