I am reading an introductory book to Sheaf theory, a kind of gentle introduction, which is nice for the most part, comparing it with other more technical books on the subject, which I am not yet prepared to follow at the moment, because of the heavy machinery usually used from category theory. But this book often omits information which I find important for my understanding of the material.
As I am understanding the Godement resolution, we define a functor $C^0: Sh(X) \to Sh(X)$ which assigns to each sheaf $F$, a flabby sheaf $C^0(F): U \mapsto \prod_{x \in U} F_{x}$ where $F_{x}$ is the stalk of $F$ at $x$. And so to get the Godement resolution, we construct a sequence $0 \to F \to C^0(F) \to C^1(F)\to...$ , where $C^{n+1}(F):= C^0(coker(C^{n-1}(F)\to C^n(F)))$. However, the part that is omited in the book is the construction of the morphisms in the sequence, which I can only guess are compositions of canonical maps. So I imagine them to be the following: $\delta_{n}:C^n(F) \to C^{n+1}(F)$ given by, for a given open set $U\subset X$, $\delta_{n,U} = i_{n,U} \circ f_{n,U} \circ \pi_{n,U}$, where $\pi_{n,U}: \Gamma(U;C^n(F)) \to \frac{\Gamma(U;C^n(F))}{im(\delta_{n-1,U})}$ is the canonical quotient map, so that $\pi$ would be only a morphism between presheaves, $f_{n,U}$ is the natural preasheaf monomorphism between the presheaf $U \mapsto \frac{\Gamma(U;C^n(F))}{im(\delta_{n-1,U})}$ to its sheafification, which is $coker (\delta_{n-1,U})$, and $i_{n,U}$ is the natural sheaf monomorphism between $coker(\delta_{n-1,U})$ and $C^0(coker(\delta_{n-1,U})) = C^{n+1}(F)$.
However, as it seems to me, these definitions not only lead to an exact sequence, but also one where it is exact as a sequence of presheaves as well, and so any sheaf $F$ would be acyclic which I know is not right.
My question is then: Where am I not thinking correctly, and also, with the correct definitions, how do we get that any flabby sheaf is acyclic? I know that $\Gamma (U;\bullet)$ is an exact functor when the first non trivial sheaf in the starting short exact sequence is flabby, but how does that translate into it being exact for the whole Godement resolution?
I appreciate any help in advance :)