= Solution
Choose an ordering of the index set $I$. The degree-$q$ <Čech cochain group> is
$$
\check C^q(\mathcal U,\mathcal F)
=\prod_{i_0<\cdots<i_q}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_q}).
$$
For $s=(s_{i_0\cdots i_q})$, the <Čech coboundary> is the alternating sum of restrictions
$$
(\delta s)_{i_0\cdots i_{q+1}}
=\sum_{j=0}^{q+1}(-1)^j
s_{i_0\cdots\widehat{i_j}\cdots i_{q+1}}
\big|_{U_{i_0}\cap\cdots\cap U_{i_{q+1}}}.
$$
The identity $\delta^2=0$ makes this the <Čech cochain complex>, and the required <Čech cohomology> is
$$
\boxed{\check H^q(\mathcal U,\mathcal F)=\ker(\delta:\check C^q\to\check C^{q+1})/\operatorname{im}(\delta:\check C^{q-1}\to\check C^q).}
$$
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