Let $R$ be a commutative ring with identity, and $M$ an $R$-module. Then $$R\otimes_R M \simeq M$$ This can be easily shown directly by construction of the isomorphism, namely $r\otimes m \mapsto rm$ for $r \in R$ and $m \in M$. Indeed, let $rm=0$. Then $r \otimes m = r\left(1\otimes m\right) = 1 \otimes rm = 1 \otimes 0 = 0$. Hence the homomorphism is injective (I skip the proof that the map in question is indeed a homomorphism; this is straight forward). On the other hand, for any $m\in M$, $1\otimes m$ gets mapped to $m$, thus showing surjectivity.
My question is, does anyone know a way to prove the same result without explicit construction of the isomorphism, i.e. via the universal property?
Consider the map $\phi: R\times M\to Z$ as a $R$-bilinear map. Also consider the natural $R$-bilinear map $\gamma: R\times M\to M$ given by $(r,m)\mapsto rm$. Note that by $R$-bilinearity $\phi(r,m)=\phi(1,rm)$.
Using these two, we construct $\psi: M\to Z$ as follows: $\psi(m)=\phi(1,m)$. Now it is easy to show that $\phi=\psi\circ \gamma$. Hence, $M$ satisfies the universal property of tensor product, so we must have $M\simeq R\otimes M$.