= Solution
The <direct image sheaf> is defined by
$$
\boxed{(\phi_*\mathcal F)(U)=\mathcal F(\phi^{-1}U)}.
$$
Its restrictions are the restrictions of $\mathcal F$. Inverse images preserve covers and intersections, so the <sheaf gluing axiom> holds immediately. The <morphism of ringed spaces> supplies
$$
\phi^\#:\mathcal O_X(U)\longrightarrow\mathcal O_Y(\phi^{-1}U).
$$
An element of $\mathcal O_X(U)$ acts on the displayed sections through this ring map and the original $\mathcal O_Y$-module action. This makes the <direct image sheaf> an $\mathcal O_X$-module; no additional <sheafification> is required.
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