Soft sheaves adapted to $f_!$ I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to $f_!$, I have to show that every acyclic complex of soft sheaves is mapped by the functor $f_!$ to an acyclic complex. 
For this, I'd say it's enough to just show that if 
$0\rightarrow \mathcal{F}\rightarrow \mathcal{G}\rightarrow\mathcal{H}\rightarrow 0$ 
is an exact sequence of soft sheaves, then $0\rightarrow \mathcal{f_!F}\rightarrow \mathcal{f_!G}\rightarrow\mathcal{f_!H}\rightarrow 0$ is also exact.
But in the book, they say the above is only enough because of an exercise from earlier which says that in an exact sequence of sheaves $0\rightarrow \mathcal{F}\rightarrow \mathcal{G}\rightarrow\mathcal{H}\rightarrow 0$ , if $\mathcal{F}$ and $\mathcal{G}$ are soft, then so is $\mathcal{H}$.
Can somebody tell me where this exercise is needed in the proof of the adaptedness of soft sheaves to $f_!$ ?

It is not needed in the proof, it just makes it easier. Because of that exercise you only need to prove that $f_!$ is right exact.
