I'm trying to do the following exercise from Vakil's notes on Algebraic Geometry:
1.3.F. EXERCISE.
Show that localization commutes with finite products, or equivalently, with finite direct sums. In other words, if $M_1, \ldots, M_n$ are $A$-modules, describe an isomorphism (of $A$-modules, and of $S^{-1}A$-modules) $S^{-1}(M_1 \times \cdots \times M_n) \to S^{-1}M_1 \times \cdots\times S^{-1}M_n$.
Show that localization commutes with arbitrary direct sums.
Show that "localization does not necessarily commute with infinite products": the obvious map $S^{-1}(M_1 \times \cdots \times M_n) \to S^{-1}M_1 \times \cdots\times S^{-1}M_n$ induced by the universal property of localization is not always an isomorphism.
The problem I'm facing is that I have an obvious map: $\frac{(m_i)}s \mapsto (\frac {m_i}s)$, but I'm not sure how to show that it is an isomorphism, in the case of finite or infinite direct sums, let alone not an isomorphism in the case of infinite products. This question basically gets at what I'm asking, but I'm finding both question and answer fairly obtuse, especially the language about "exact functors" about which I know next to nothing.