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) $A\otimes M\cong M$