It is known that the tensor product is characterized by its universality in the category of $A$-modules.
Does the following proposition hold?
Proposition There exists only one operation $\otimes$ that satisfies the following four properties in the category of $A$-modules.
Let $M$, $N$, and $P$ be $A$-modules. Then
(i) $M\otimes N\cong N\otimes M$
(ii) $(M\otimes N)\otimes P\cong M\otimes (N\otimes P)$
(iii) $(M\oplus N)\otimes P\cong (M\otimes P)\oplus (N\otimes P)$
(iv) $A\otimes M\cong M$