This is a follow-up to my previous question.
Question:
How are they able to replace $A \otimes M$, $B \otimes M$ and $C \otimes M$ by $A \times M$, $B \times M$ and $C \times M$ respectively in the commutative diagram?
I'm especially confused here since the tensor product doesn't just consist of simple tensors. It's possible that some weird linear combination $\sum_i a_i \otimes m_i$ lies in $\ker(j \otimes 1)$ but the individual simple tensors $a_i \otimes m_i$ don't, where each $a_i \in A, m_i \in M$.