= Solution
The intersection hypothesis says that $X$ is a <semi-separated scheme>. For an <affine open subscheme> $V\subseteq X$, the intersection $V\cap U$ is affine. Restricting the given <short exact sequence of sheaves> to it and applying the <affine module-sheaf equivalence> gives an <exact sequence> of <global sections>:
$$
0\longrightarrow\Gamma(V\cap U,\mathcal M')
\longrightarrow\Gamma(V\cap U,\mathcal M)
\longrightarrow\Gamma(V\cap U,\mathcal M'')\longrightarrow0.
$$
These are exactly the sections on $V$ of the three <direct image sheaves>. Affine opens form a basis, and every section of the last sheaf on such a basis open lifts to the middle sheaf. This proves surjectivity as a <sheaf morphism>; left exactness of <direct image> supplies the other positions. Hence \b[direct image preserves this short exact sequence]. The crucial ingredient is exactness of sections of <quasi-coherent sheaves> on an affine intersection; arbitrary <open immersions> need not have this property.
For the cohomology comparison, every nonempty finite intersection
$$
U_{i_0\cdots i_p}=U_{i_0}\cap\cdots\cap U_{i_p}
$$
is affine. This follows by <mathematical induction>, intersecting the affine intersection already obtained with the next affine open. The restriction of $\mathcal F$ to it is quasi-coherent, so <vanishing of quasi-coherent cohomology on an affine scheme> gives
$$
H^q(U_{i_0\cdots i_p},\mathcal F)=0\qquad(q>0).
$$
We now prove why this local vanishing gives the <acyclic cover theorem>, rather than identifying the two sorts of cohomology without a comparison.
Take a <flasque resolution> $0\to\mathcal F\to\mathcal I^0\to\mathcal I^1\to\cdots$. Here a <flabby sheaf> has surjective restriction maps; its restrictions to open subsets are still flabby and have zero higher <sheaf cohomology>. Form the <double complex>
$$
C^{p,q}=\prod_{i_0<\cdots<i_p}
\Gamma(U_{i_0\cdots i_p},\mathcal I^q).
$$
The horizontal differential is the alternating restriction map $d_{\mathrm C}$; the vertical one $d_{\mathrm I}$ is induced by the resolution. They commute, so the total differential in bidegree $(p,q)$ is $d_{\mathrm C}+(-1)^p d_{\mathrm I}$ and has square zero. The <Čech resolution on a semi-separated scheme> uses precisely these intersections.
A <flabby sheaf> has zero positive <Čech cohomology> for a finite <open cover>, and its degree-zero <Čech cohomology> is its <global sections>. One way to establish this auxiliary fact is to use the exact augmented two-open complex
$$
0\to\Gamma(V\cup W,\mathcal I)
\to\Gamma(V,\mathcal I)\oplus\Gamma(W,\mathcal I)
\to\Gamma(V\cap W,\mathcal I)\to0.
$$
Exactness at the first two terms is the <sheaf gluing axiom>; the last map is onto because a section on the intersection extends to $V$. For the induction step, write $V$ for the union of all but the last open and $W$ for the last open. Separate Čech cochains according to whether their index list contains the last index. The resulting two-block complex compares the smaller cover of $V$ with its restricted cover of $V\cap W$, together with $\Gamma(W,\mathcal I)$ in degree zero. By induction those two smaller cover complexes have cohomology only in degree zero, where they give $\Gamma(V,\mathcal I)$ and $\Gamma(V\cap W,\mathcal I)$. The displayed two-open exact sequence then gives zero positive-degree cohomology for the full cover. This proves the auxiliary fact by induction on the number of opens. Thus horizontal cohomology of $C^{\bullet,q}$ consists only of $\Gamma(X,\mathcal I^q)$ in degree zero. Computing the cohomology of the total complex first horizontally therefore gives $H^n(X,\mathcal F)$, by the <resolution principle for sheaf cohomology>.
On the other hand, vertical cohomology is
$$
H^q(C^{p,\bullet})
=\prod_{i_0<\cdots<i_p}H^q(U_{i_0\cdots i_p},\mathcal F),
$$
since the restricted <flasque resolutions> compute cohomology on each intersection. The already established affine vanishing makes all rows with $q>0$ zero. The surviving row $q=0$ is exactly the <Čech cochain complex> $\check C^\bullet(\mathcal U,\mathcal F)$. Computing total cohomology first vertically therefore gives $\check H^n(\mathcal U,\mathcal F)$. These two computations are justified by the two filtrations of the first-quadrant <double complex>: in every total degree only finitely many terms occur, and the cover also bounds the horizontal degree. Their edge maps give the natural identification
$$
\boxed{\check H^p(\mathcal U,\mathcal F)\cong H^p(X,\mathcal F)\quad(p\geq0).}
$$
For negative degrees both groups are zero by convention. In particular, degree zero is the usual identification by the <sheaf gluing axiom>, not merely a comparison of dimensions.
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