I am confused about notation concerning tensor products of sheaves of modules. I know that given a ringed space $X$ and $\mathcal{O}_X$-Modules $\mathcal{F}$ and $\mathcal{G}$ their tensor product is defined to be the associated sheaf to the presheaf $U \mapsto \mathcal{F}(U) \otimes_{\mathcal{O}_X(U)} \mathcal{G}(U)$. Despite the sheafification process I have seen authors writing sections in the form $a \otimes b$.
In the case I am dealing with at the moment, a morphism of $\mathcal{O}_X$-Algebras $\mathcal{O}_X[T, T^{-1}] \to \mathcal{O}_X[T, T^{-1}] \otimes \mathcal{O}_X[T, T^{-1}]$ is defined by $T \mapsto T \otimes T$ and I cannot see why this is meaningful, to be precise: why $T \otimes T$ represents a global section of the target. Is there some kind of convention or abuse of notation I am missing out on? Any help will be appreciated.