 Solution

The direct image sheaf is defined byIts 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 suppliesAn 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.