Tensor product commutes with direct limits (a particular question) Let $(\{M_n\}, \{f_{nm}\})$ be a direct system of right $R$-modules and $N$ be a left $R$-module. We have:
$$\left( \lim_{\rightarrow}M_n \right) \otimes N \cong \lim_{\rightarrow}(M_n \otimes N) $$
What I would like to do is construct a forward morphism and a backward morphism and prove that they are inverses.
For the forward map, I first took the map $$f: \left( \lim_{\rightarrow}M_n \right) \times N \longrightarrow\lim_{\rightarrow}(M_n \otimes N) $$
given by $f([x_n], y) = [x_n \otimes y]$. It is easy to check that this is a balanced product. By the universal mapping property, it induces a morphism $$\left( \lim_{\rightarrow} M_n \right) \otimes N \to \lim_{\rightarrow}(M_n \otimes N) $$
For the backward morphism, I thought about the following:
$$\varphi \left( \left[ \sum_{t=1}^{r} m_n^{(t)} \otimes y^{(t)} \right] \right) = \sum_{t=1}^s \left[m_n^{(t)}\right] \otimes y^{(t)}$$
However, I am not sure how to prove that this map is well-defined (well, is it?)
If this map works as the needed backward morphism, please let me know how to prove that it's well-defined. If it isn't, I would appreciate it if you can point me in the right direction (please no hom's, Yoneda and whatnot)
 A: To construct a map from a direct limit, it's often easiest to use the universal property of the direct limit by producing maps from each of the terms in the direct limit that respect the maps between the terms. 
For example, let $M=\displaystyle\lim_{\to} M_n$, and with ${\iota}_{n} : M_n\to M$ the corresponding maps. Then $\iota_n \otimes \text{id}_N : M_n\otimes N \to M\otimes N$ respect the maps $f_{nm}\otimes \text{id}_N$, since the maps $\iota_n$ respect the $f_{nm}$. Thus there is an induced map $\lim\limits_{\to} (M_n\otimes N) \to M\otimes N$. 
It's typically better to construct maps using the universal properties than by defining them on elements when working with limits, colimits, and tensor products, because then the maps are certainly well-defined, whereas its difficult to show that a map defined on elements is well-defined.
Oh and I guess to answer your question, to show its well defined, you'd have to go back to the construction of the direct limit, define your map on the direct sum and show that your map respects the relations of the direct limit.
