I seem to encounter this issue whenever a question involves quotient objects. In this case, I have modules $M_1$ and $M_2$ and subsets $N_1$ and $N_2$ thereof respectively. It is given that $N_1$ and $N_2$ are submodules, thus so too is their set-theoretic Cartesian product, namely $N_1 \times N_2$. Thus, all expressions in the formula below are well-defined. The task is to show that $$\frac{M_1 \times M_2}{N_1 \times N_2} \simeq \frac{M_1}{N_1} \times \frac{M_2}{N_2}.$$
As a first step, I'd like to show that there exists a function $$f : \frac{M_1 \times M_2}{N_1 \times N_2} \rightarrow \frac{M_1}{N_1} \times \frac{M_2}{N_2}$$ such that for all $(m_1,m_2) \in M_1 \times M_2$ it holds that
$$f((m_1,m_2)+N_1 \times N_2) = (m_1+N_1, m_2 + N_2),$$
but I can't work out how to do it. Since functions are generalized by relations, one approach would be to define a relation with the right properties and then prove that it is a function. But I can't even work out the appropriate relation to define.
Help, anyone?