Index notation in question about distribuitivity of tensor product over direct sum

Thanks to @peek-a-boo I edited my question

In this post Tensor product and direct sum the author has the map $\varphi : (\bigoplus_\alpha M_\alpha)\times N\to \bigoplus_\alpha (M_\alpha\otimes N)$

defined by $$\varphi((m_\alpha), n)=(m_\alpha\otimes n)$$

Suppose $m_\alpha=(m_1,...,m_k)$ and $n$ is an element of $N$.

Is an element $x \in \bigoplus_\alpha (M_\alpha\otimes N)$ of the form $x=(m_1,...,m_k)\otimes n$ or of the form $x=(m_1\otimes n,...,m_k\otimes n)$?

No, an element of the (external) direct sum $\bigoplus\limits_{\alpha\in A}M_{\alpha}$ is by definition a function $f:A\to\bigcup_{\alpha\in A}M_{\alpha}$ such that for each $\alpha\in A$, we have $f(\alpha)\in M_{\alpha}$, and such that $\{\alpha\in A\,: f(\alpha)\neq 0\}$ is a finite set (i.e finitely supported elements of the cartesian/direct product $\prod\limits_{\alpha\in A}M_{\alpha}$).

It is usually tradition to not use the function notation as in $f$. Rather, one displays the function by writing out its values as in $(f(\alpha))_{\alpha\in A}$. Since the letter $f$ isn't nice, we typically write this as $(m_{\alpha})_{\alpha\in A}$ to mean an element of $\bigoplus\limits_{\alpha\in A}M_{\alpha}$. So, once again, for each $\alpha\in A$, we have $m_{\alpha}\in M_{\alpha}$.

Finally, the index set $A$ is usually understood by everyone based on context, so we omit that from the notation as well, and simply write $(m_{\alpha})$ (the round brackets is emphasizing that we have a collection of elements) to mean an element of $\bigoplus_{\alpha}M_{\alpha}$.

Btw when the index set $A$ is finite, we can identify it with $\{1,\dots, k\}$ for some $k\in\Bbb{N}$, in which case we write an element of the direct sum (or equivalently direct/cartesian product) simply as $(m_1,\dots, m_k)$.

This is just like how for sequences of real numbers, we usually denote them as $(x_n)_{n\in \Bbb{N}}$ or $(x_n)_{n=1}^{\infty}$ or $\{x_n\}_{n=1}^{\infty}$, rather than the single letter denoting the function $x:\Bbb{N}\to\Bbb{R}$.