Let $f:X \rightarrow Y$ be a finite morphism of locally Noetherian schemes. If $\mathcal{F}$ is an $f_*\mathcal{O}_X$-module, why is $\mathcal{F}$ endowed with the structure of an $\mathcal{O}_X$-module?
This should be really basic, but I struggle to see this. For each open subset $U \subset Y$ we can multiply $\mathcal{F}(U)$ by scalars in $\mathcal{O}_X(f^{-1}(U))$, thus $\mathcal{F}$ gives us for each open subset in $Y$ an $\mathcal{O}_X$-module, but not for each open subset of $X$...
Why is $\mathcal{F}$ endowed with the structure of an $\mathcal{O}_X$-module?