I am looking for a certain way of proving the following :
Let $f. X \rightarrow S$ be a morphism of schemes. Suppose that f is quasiseparated and quasicompact, or that X is noetherian. Let $\mathcal{G}$ be a locally free sheaf on S and $\mathcal{F}$ a quasicoherent sheaf on X. Show that we have a canonical isomorphism: $$f_\ast \mathcal{F} \otimes_{\mathcal{O}_S} \mathcal{G} \cong f_\ast (\mathcal{F} \otimes_{\mathcal{O}_X} f^\ast \mathcal{G}).$$
I an prove this by using a gluing argument on affines , but I would want a proof that is more global, but finishes it off with checking the isomorphism on stalks Let me show you what I mean. We have a morphism $$\alpha: f_\ast \mathcal{F} \otimes_{\mathcal{O}_S} \mathcal{G} \rightarrow f_\ast \mathcal{F} \otimes_{\mathcal{O}_S} f_\ast f^\ast \mathcal{G}$$ given by tensoring the canonical morphism $\eta: \mathcal{G} \rightarrow f_\ast f^\ast \mathcal{G}$ with the identity on $f_\ast \mathcal{F}$. It can be shown that we also have a canonical morphism: $$\beta: f_\ast \mathcal{F} \otimes_{\mathcal{O}_S} f_\ast f^\ast \mathcal{G} \rightarrow f_\ast (\mathcal{F} \otimes_{\mathcal{O}_X} f^\ast \mathcal{G})$$ and composing $\alpha$ and $\beta$ we get our desired morphism. Is it, by only this, possible to check this isomorphism on the stalks? If so - how could we do it? What properties do we use of the stalks of $f_\ast (\mathcal{F} \otimes_{\mathcal{O}_X} f^\ast \mathcal{G})$ in that case?
Thanks for help!