I am trying to show that tensor products and directs sums commute, i.e. $$ M \otimes \left( \bigoplus_{i \in I} N_i \right) \simeq \bigoplus_{i \in I} (M \otimes N_i). $$
I tried to consider the well-defined homomorphism $$ \left( m, (n_i) \right) \mapsto (m \otimes n_i)_i. $$
On this page (proof of Lemma 8.12), it is said that this map is obviously surjective. But this is far from being obvious to me. An arbitrary element of $\bigoplus_{i \in I} (M \otimes N_i)$ has the form $(m_i \otimes n_i)_i$ and not $(m \otimes n_i)_i$ as one might think. This makes it more difficult to find an element of $M \otimes \left( \bigoplus_{i \in I} N_i \right)$ which maps to that element.
Could you please explain this "obvious" observation to me? Thank you!