Cech cohomology computes quasi-coherent cohomology on a separated scheme

Let $U=\mathrm{Spec}B$, $V=\mathrm{Spec}C$ be affine opens of $X$. Fix an affine open $W$ of the base of $X$ over which both lie — for the structure morphism $X\to \mathrm{Spec}\mathbb{Z}$ one may take $W=\mathrm{Spec}\mathbb{Z}$ itself. Since $X$ is separated, the criterion [F1] applies to the pair $U,V$ and shows that $U\cap V$ is affine, the empty intersection being the empty affine scheme.

Every finite intersection of members of the cover is affine. The claim is clear for one member; if $U_{i_1}\cap \cdots \cap U_{i_k}$ is affine, its intersection with the affine open $U_{i_{k+1}}$ is affine by step 1.1. Consequently, for every nonempty finite intersection $W$ of members of the cover, $W$ is affine and $\mathcal{F}{\mathrm{\mid }}_W$ is quasi-coherent by [F2], so $H^q(W,\mathcal{F}{\mathrm{\mid }}_W)=0$ for every $q>0$ by [F3]. This is exactly the acyclicity condition of [F4] for $\mathcal{F}$ and the cover, whose index set $\{0,\dots ,r\}$ is linearly ordered by the given ordering of the cover.

Apply [F5] to the linearly ordered $\mathcal{F}$-acyclic cover $U_0,\dots ,U_r$: the canonical comparison map ${\check{H}}^q(\mathcal{U},\mathcal{F})\to H^q(X,\mathcal{F})$ is an isomorphism for every $q\ge 0$, where the left-hand side is the cohomology of the ordered complex $C^{\bullet }(\mathcal{U},\mathcal{F})$ by [F6]. Boundary cases: if $X=\mathrm{\varnothing }$ every member of the cover is empty (including when the index set is empty), so every cochain group and both sides vanish; if $r=0$ the complex has the single term $\mathrm{\Gamma }(X,\mathcal{F})$ and the comparison is the identity in degree zero; degree $q=0$ is the global-sections identification and positive $q$ is covered by the acyclicity of step 2.1. The Axiom of Choice is inherited from [F3] and [F5], with the finite induction of step 2.1 making no further selection.