Let $\mathcal{F}$ be a sheaf of abelian groups over a topological space $X$ with an open covering $\mathcal{U}=\{U_i\}_{i\in I}$. The Čech complex of $\mathcal{F}$ over $\mathcal{U}$ is denoted as $\check{\mathcal{C}}^{\bullet}(\mathcal{U},\mathcal{F})$. The alternating Čech complex of $\mathcal{F}$ over $\mathcal{U}$ is denoted as $\check{\mathcal{C}}^{\bullet}_{alt}(\mathcal{U},\mathcal{F})$. Let $i:\check{\mathcal{C}}^{\bullet}_{alt}(\mathcal{U},\mathcal{F})\hookrightarrow \check{\mathcal{C}}^{\bullet}(\mathcal{U},\mathcal{F})$ be the canonical injection. Then it induces a map on the cohomology groups: $$H^q(i):H^q(\check{\mathcal{C}}^{\bullet}_{alt}(\mathcal{U},\mathcal{F}))\to H^q(\check{\mathcal{C}}^{\bullet}(\mathcal{U},\mathcal{F}))$$
It's known that $H^q(i)$ is an isomorphism for all $q\geq 0$. But all the proofs (e.g. tag 01FM) I found is based on the assumption that there exists a total order (weaker than well order) on $I$.
If we assumed Axiom of Choice (equivalent to Well-ordering Theorem), then the proof is fine. But can we prove it without AC?
Injectivity of $H^q(i)$ is trivial for all $q$. Surjectivity for the cases $q=0,1$ are trivial. But I got stuck at $q=2$.
