= Solution
The <direct image sheaf> is defined on an <open set> $V\subseteq Y$ by
$$
(\phi_*\mathcal F)(V)=\mathcal F(\phi^{-1}V).
$$
Restrictions are those of $\mathcal F$, and the <sheaf gluing axiom> follows by taking inverse images of an <open cover>. The <morphism of ringed spaces> supplies $\phi^\#: \mathcal O_Y\to\phi_*\mathcal O_X$. Thus $a\in\mathcal O_Y(V)$ acts on $s\in\mathcal F(\phi^{-1}V)$ by $\phi^\#(a)s$, making $\phi_*\mathcal F$ a <sheaf of modules> over $\mathcal O_Y$. This definition uses no <quasi-coherence>.
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