Godement terms are flasque and compute cohomology

For an open $V\subseteq X$ the assignment $V\mapsto {\prod }_{x\in V}{\mathcal{E}}_x$ with the evident projections as restriction maps is a sheaf: locality and gluing are checked coordinate by coordinate in the product of groups, using [F4] at each point, and by [F2] it is the product of the skyscraper sheaves $i_{x,\ast }{\mathcal{E}}_x$ in $\mathrm{A}\mathrm{b}(X)$. Moreover the germ maps $\mathcal{E}(V)\to {\prod }_{x\in V}{\mathcal{E}}_x$, $s\mapsto (s_x{)}_{x\in V}$, are compatible with restrictions by the second half of [F1], so they define a morphism ${\epsilon }_{\mathcal{E}}:\mathcal{E}\to C^0(\mathcal{E})$ [F1].

For every abelian sheaf $\mathcal{E}$ on $X$ the germ map ${\epsilon }_{\mathcal{E}}$ is injective: if $s\in \mathcal{E}(V)$ maps to $0$, all germs $s_x$ with $x\in V$ vanish, so $s=0$ by [F1]; hence $\ker ({\epsilon }_{\mathcal{E}})=0$ and ${\epsilon }_{\mathcal{E}}$ is a monomorphism in the abelian category $\mathrm{A}\mathrm{b}(X)$ [F3]. Consequently the cokernel $Q(\mathcal{E})=\mathrm{coker}({\epsilon }_{\mathcal{E}})$ fits into a short exact sequence $0\to \mathcal{E}\to C^0(\mathcal{E})\to Q(\mathcal{E})\to 0$ in $\mathrm{A}\mathrm{b}(X)$ [F3].

Every term $C^0(\mathcal{E})$ is flasque: for open $U\subseteq V$ the restriction map ${\prod }_{x\in V}{\mathcal{E}}_x\to {\prod }_{x\in U}{\mathcal{E}}_x$ is the projection on the coordinates in $U$ [F2, step 1.1], which is surjective. Hence each term $C^n(\mathcal{F})=C^0(Q^{n-1}(\mathcal{F}))$ of the Godement resolution is flasque, which is assertion 1.

The coaugmented complex is exact. At $\mathcal{F}$ this is the injectivity of ${\epsilon }_{\mathcal{F}}$ from step 1.2. In degree $n\ge 0$ write $q_n:C^n(\mathcal{F})\to Q^n(\mathcal{F})$ for the quotient morphism, so that $d^n={\epsilon }_{Q^n}\circ q_n$ by the definition of the Godement differential and $q_n$ is the cokernel of the monomorphism ${\epsilon }_{Q^{n-1}}$ of step 1.2; by the exactness of $0\to Q^{n-1}(\mathcal{F})\to C^n(\mathcal{F})\stackrel{q_n}{\to }{Q}^n(\mathcal{F})\to 0$ in $\mathrm{A}\mathrm{b}(X)$ [F3] one has $\mathrm{im}(d^{n-1})=\mathrm{im}({\epsilon }_{Q^{n-1}})=\ker (q_n)$, while $\ker (d^n)=\ker ({\epsilon }_{Q^n}\circ q_n)=\ker (q_n)$ because ${\epsilon }_{Q^n}$ is a monomorphism by step 1.2. Hence $\mathrm{im}(d^{n-1})=\ker (d^n)$ and the complex is exact at every term, which is assertion 2. [F3, step 1.2, given]

By [F7] AC gives DC. Each term $C^n(\mathcal{F})$ is flasque by step 2.1, hence $\mathrm{\Gamma }$-acyclic on $X$ by [F5]; the successive cokernels of the resolution are $Z^0=\mathcal{F}$ and $Z^{q+1}=Q^q(\mathcal{F})$ for $q\ge 0$, which are abelian sheaves on $X$, hence lie in the domain of the supplied injective resolution datum by [F8]. Applying [F6] to the additive left exact functor $\mathrm{\Gamma }(X,-)$ (Global sections of an abelian sheaf), the datum and the acyclic resolution $0\to \mathcal{F}\to C^{\bullet }(\mathcal{F})$ gives $H^q(X,\mathcal{F})=R^q\mathrm{\Gamma }(X,\mathcal{F})\cong H^q(\mathrm{\Gamma }(X,C^{\bullet }(\mathcal{F})))$ for every $q\ge 0$, with the naturality supplied by the functoriality of the Godement construction recorded in Godement resolution of an abelian sheaf. This is assertion 3. ∎ [F5, F6, F7, F8, step 2.1, step 2.2]