Consider the category of chain complexes (of say abelian groups) that vanish in negative indexes.
We know there is a long exact sequence of complexes which vanish in negative indexes for a ses $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow0$:
$..H_n(A)\to H_n(B)\to H_n(C)\to H_{n-1}(A)\to ...H_0(A)\to H_0(B)\to H_0(C) \rightarrow 0$.
I really wanted to see this a special case of left derived functors, but I can't find a way to do this.
Here is the problem-
We'd like to take the functor $H_0$, but it's only right exact if we work in the category of complexes which vanish in negative degrees. But then, naively we don't have projectives.
Possible solutions- Maybe I'm just wrong and there are enough projectives?
Another possible weird solution is something that is not left or right derived functors, but something which generalizes them which will give us the general long exact sequence between three complexes? Like maybe if a category has enough projectives AND enough injectives we can somehow take a long exact sequence?