Given a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ over $X=\text{Proj}S$ for a graded ring $S$, I'd like to precisely understand how one can give $S$-module structure to the direct sum $$\underset{m\in\mathbb{Z}}\bigoplus\Gamma(X,\mathcal{F}(m)).$$
More precisely looking at here: https://stacks.math.columbia.edu/tag/01MM, I know that there is a natural map $S\to\bigoplus\Gamma(X,\mathcal{O}(n))$ given by mapping $s\in S_d$ to the gluing of $\displaystyle\frac{s}{1}\in\mathcal{O}(d)(D(f))$ for an open cover made of $D(f)$'s. Moreover I also understand the construction of the morphism $\mathcal{O}(n)\otimes_{\mathcal{O}}\mathcal{O}(m)\to\mathcal{O}(n+m)$, and hence, by tensoring, of the map $\mathcal{O}(n)\otimes_{\mathcal{O}}{\mathcal{F}(m)}\to\mathcal{F}(n+m)$.
My question is: why one can say that we have a morphism $$\underset{n\in\mathbb{N}}\bigoplus\Gamma(X,\mathcal{O}(n))\times\underset{m\in\mathbb{Z}}\bigoplus\Gamma(X,\mathcal{F}(m))\to\underset{m\in\mathbb{Z}}\bigoplus\Gamma(X,\mathcal{F}(m))?$$
This would give a $S$-module structure by what said before, but I can't understand how to find an element in $\mathcal{F}(n+m)(X)$ given $(s,t)\in\mathcal{O}(n)(X)\times\mathcal{F}(m)(X)$ using the morphism $\mathcal{O}(n)\otimes_{\mathcal{O}}{\mathcal{F}(m)}\to\mathcal{F}(n+m)$.
The main problem seems to be that for what I know it is not true in general that $$\mathcal{O}(n)(X)\otimes_{\mathcal{O}(X)}\mathcal{F}(m)(X)\cong(\mathcal{O}(n)\otimes_{\mathcal{O}}\mathcal{F}(m))(X),$$ otherwise I would simply consider the element $s\otimes t $. What am I missing?
Thanks in advance!