= Solution
Form the <presheaf> whose value on $U\subseteq X$ is
$$
\mathcal G(U)\otimes_{\mathcal O_X(U)}\mathcal H(U),
$$
with restrictions induced from those of the two <sheaves of modules> and the <structure sheaf>. Its <sheafification> is the <tensor product of sheaves> $\mathcal G\otimes_{\mathcal O_X}\mathcal H$. Sheafification is needed because taking a <tensor product> on each <open set> need not itself satisfy the gluing axiom. The resulting <sheaf> represents bilinear <sheaf> morphisms, and <localization> gives the <stalk> formula
$$
\boxed{(\mathcal G\otimes_{\mathcal O_X}\mathcal H)_P
\cong\mathcal G_P\otimes_{\mathcal O_{X,P}}\mathcal H_P}.
$$
Indeed, <sheafification> preserves <stalks>, and filtered <direct limits> commute with the <module> <tensor product>.
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