Let $K^{\bullet}$ be a bounded complex of abelian étale sheaves on a quasi-compact and quasi-separated scheme $X$.

For any étale cover $\mathcal{U} :=\{ U_i\to X\}_{i\in I}$, can we find a refinement $\mathcal{U}'$ of $\mathcal{U}$ with $\mathcal{U}'$ finite (the answer to this should be easily yes, but I just want to double check)

We have a Cech-to-cohomology spectral sequence

$$E_2^{p,q} := H^p(\text{Tot}(\check{C}(\mathcal{U},\underline{H}^q(K^{\bullet})))\Rightarrow H^{p+q}(X_{\rm ét},K^{\bullet})$$ for any $\mathcal{U}$.

I must be misunderstanding the meaning of convergence of this spectral sequence for **any** $\mathcal{U}$. Zariski covers are, in particular, étale covers. But if I choose $\mathcal{U}$ a Zariski cover, the spectral sequence should converge to **Zariski** hypercohomology of $K^{\bullet}$, which won't be **étale** hypercohomology.

What is the meaning of **for any $\mathcal{U}$**, then?

Thanks a lot.