Let $X$ a topological space (ie a scheme) and $E$ be a sheaf of abelian groups on $X$. Now we can associate to $X and E$ sheaf cohomology groups $H^i(X,E)$. For an explicit construction choose an injective resolution
$$ 0 \to E \to I_0 \to I_1 \to I_2 \to ... $$
Then the sheaf cohomology groups are cohomology groups of the chain complex
$$ 0 \to I_0(X) \to I_1(X) \to I_2(X) \to ... $$
for global sections $I_j(X)$ of $I_j$. In some cases the Čech cohomology gives a good approximation to sheaf cohomology that is often useful for computations. If $\mathcal{U}$ is an open cover of $X$, then Čech cohomology $H^j(\mathcal{U}, E)$ is defined as the cohomology of an explicit complex of abelian groups with $j$th group
$$ C^i(\mathcal{U}, E) = \prod_{i_0 < ... <i_j}E(U_{i_0} \cap ... \cap U_{i_j}) .$$
The cohomology groups $\check{H}^{j}(X,E)$ are defined as the direct limit of the $H^i(U, E)$ over all open covers $U $ of $X$. In literature is often noted that there is a comparison map
$$ \check{H}^{j}(X,E) \to H^j(X,E) $$
which in some nice situation is an isomorphism. But I haven't found somewhere an explanation how this comparison map is explicitely constructed. Could somebody give a sketch how the construction works? Obviously it is sufficient to construct the maps $ H^j(\mathcal{U}, E)\to H^j(X, E)$ and then push it to direct limit. How the last map is constructed?