Browsing over some questions, I found that the natural homomorphism from $(\prod M_i)\otimes N\to \prod(M_i\otimes N)$ is given by $(\prod m_i)\otimes n\mapsto \prod(m_i\otimes n)$.
This of course seems very natural, but how does one know it is in fact well defined? The infinite product of representative from $M$ is giving me a difficult time accepting this, although I don't doubt it to be true. Thanks.