The one-member cover of the circle has no Čech H1, but the sheaf H1 is nonzero

Take $X=S^1$ the circle with quotient map $p$ [F9], the index set $I=\{0\}$ with its unique linear order and $U_0:=S^1$; the circle is open in itself, so $\mathcal{U}=(U_0)$ is a cover of $X=S^1$ by open subsets. The increasing tuples in $I$ are the single $0$-tuple $(0)$ in degree $0$, and there is no increasing $(p+1)$-tuple for $p\ge 1$; hence by [F1] $C^0(\mathcal{U},\underset{‾}{\mathbb{Z}})=\underset{‾}{\mathbb{Z}}(U_0)=\underset{‾}{\mathbb{Z}}(S^1),C^p(\mathcal{U},\underset{‾}{\mathbb{Z}})=0(p\ge 1)\mathrm{.}$ So the differential ${\delta }^0:C^0\to C^1$ has target $0$ and is the zero map, and all higher differentials vanish as well. By [F2] the cohomology of the complex $C^0\to 0\to \cdots$ is ${\check{H}}^0(\mathcal{U},\underset{‾}{\mathbb{Z}})=\ker {\delta }^0=C^0=\underset{‾}{\mathbb{Z}}(S^1)$ and ${\check{H}}^p(\mathcal{U},\underset{‾}{\mathbb{Z}})=\ker {\delta }^p\mathrm{/}\mathrm{im}{\delta }^{p-1}=0\mathrm{/}0=0$ for every $p\ge 1$; in particular ${\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})=0$.

By [F4] there is a global section $q$ of the quotient sheaf $\mathcal{Q}$ on $S^1$ — the sheafification of the presheaf quotient of the continuous real-valued functions by the locally constant integer-valued functions — whose connecting class $\mathrm{\partial }(q)\in H^1(S^1,\underset{‾}{\mathbb{Z}})$ is nonzero; hence $H^1(S^1,\underset{‾}{\mathbb{Z}})\mathrm{\ne }0$. The coefficient sheaf there is the constant sheaf with value $\mathbb{Z}$ on $S^1$, identified with the subsheaf of locally constant integer-valued functions [F8], which is the same sheaf $\underset{‾}{\mathbb{Z}}$ used in [step 1.1]; the two occurrences of $H^1(S^1,\underset{‾}{\mathbb{Z}})$ denote the same group, formed from the same fixed injective resolution datum.

By [F3], under the Axiom of Choice, the comparison map for the cover $\mathcal{U}$ of [step 1.1] is defined: ${\phi }_{\mathcal{U}}^1:{\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})⟶H^1(S^1,\underset{‾}{\mathbb{Z}})\mathrm{.}$ Its source is the zero group by [step 1.1], so ${\phi }_{\mathcal{U}}^1$ is the zero homomorphism, while its target is nonzero by [step 2.1]. A homomorphism whose target is nonzero and whose source is the zero group has image $0\mathrm{\ne }{H}^1(S^1,\underset{‾}{\mathbb{Z}})$, so it is not surjective and in particular not an isomorphism.

The cover $\mathcal{U}$ is not $\underset{‾}{\mathbb{Z}}$-acyclic. Indeed, by [F6] $\mathcal{U}$ is $\underset{‾}{\mathbb{Z}}$-acyclic exactly when $H^q(W,\underset{‾}{\mathbb{Z}}{\mathrm{\mid }}_W)=0$ for every $q>0$ and every nonempty finite intersection $W$ of members of $\mathcal{U}$; the only member is $U_0=S^1$, so the only such $W$ is $S^1$ itself, whose restriction of $\underset{‾}{\mathbb{Z}}$ is $\underset{‾}{\mathbb{Z}}$, and $H^1(S^1,\underset{‾}{\mathbb{Z}})\mathrm{\ne }0$ by [step 2.1]. Hence the hypothesis of the Leray comparison theorem [F7] fails for this cover, and no contradiction with that theorem arises from [step 3.1]: the theorem gives an isomorphism only for $\mathcal{F}$-acyclic covers, and the one-member cover of the circle is not one.

Assembling the pieces for the circle, the constant sheaf $\underset{‾}{\mathbb{Z}}$ and the one-member cover $\{S^1\}$: ${\check{H}}^1(\mathcal{U},\underset{‾}{\mathbb{Z}})=0$ by [step 1.1], the comparison map ${\phi }_{\mathcal{U}}^1$ is the zero map out of the zero group by [step 3.1], the sheaf cohomology $H^1(S^1,\underset{‾}{\mathbb{Z}})$ is nonzero by [step 2.1], and the cover fails acyclicity by [step 4.1]. Therefore the assertion that the comparison map ${\phi }_{\mathcal{U}}^p$ is an isomorphism in every degree for every cover is false, with degree $p=1$ and this cover as the witness; on the other hand ${\phi }_{\mathcal{U}}^0$ is an isomorphism for every cover, since it is the identity of the global sections up to the canonical identifications, so degree zero provides no obstruction. The Axiom of Choice of [F5] enters exactly twice: in [step 3.1] through the construction of the comparison map, which uses the Godement resolution of [F3], and in [step 2.1] through the long exact sequence in sheaf cohomology of [F4], whose connecting homomorphism is formed from injective resolutions; no other selection is used, the cochains of [step 1.1] being a single section group. ∎