In my algebraic geometry class, we defined sheaf cohomology using flabby sheaves, and the functor on the category of sheaves on a space $X$:
$$
D: \mathcal F \mapsto D\mathcal F
$$
where
$$
D\mathcal F(U)=\left\{s:U\to \bigsqcup_{p\in X}\mathcal F_p\middle| s(p)\in\mathcal F_p\right\}
$$
$D\mathcal F$ is then a flabby sheaf containing $\mathcal F$.
But when I did some further research into derived functor cohomology, it became clear that the important property you need is that sheaf categories 'have enough injectives' – so every sheaf is a subsheaf of some injective sheaf. At first I thought that flabby sheaves and injective sheaves were the same thing, but then I found out that in fact, being injective is a stronger property: every injective sheaf is flabby, but not every flabby sheaf is injective.
How then, were we able to develop the cohomology theory using flabby sheaves, when one in general needs to consider injective objects?
Best Answer
That is true, not every flabby sheaf is injective, but the important thing to compute cohomology is to be able to build an acyclic resolution. Injectives are acyclic, but they are not the only class of acyclic objects! In fact, flabby sheaves are acyclic as well (for the $\Gamma(X,-)$ functor). This of course would not be enough if you were not able to construct flabby resolutions for every sheaf. But luckily, every sheaf has a canonical flabby resolution, called the Godement resolution. The functor you describe gives you the first piece $0\to \mathcal F\to D\mathcal F$ of the resolution.