Here I will use the universal property of tensor products regarding bilinear maps to prove that the construction of tensor products is associative. For simplicity, I will take the ring $R$ to be commutative so that we can view $R$-modules as $(R,R)$-bimodules.

Theorem: Let $R$ be a commutative and unitary ring, $M, N$ and $L$ be $R$-modules. Then there is a unique isomorphism

$(M \otimes N) \otimes L \simeq M \otimes (N \otimes L).$

Proof: For a fixed $l,$ it is easy to see that the map $\varphi_l : M \times N \to M \otimes (N \otimes L)$ such that $\varphi_l(m,n) = m \otimes (n \otimes l)$ is bilinear. For example we have $\varphi_l (m, n_1 + n_2) = m \otimes (n_1 \otimes l + n_2 \otimes l) = m \otimes (n_1 \otimes l) + m \otimes (n_2 \otimes l) = \varphi_l(m, n_1) + \varphi_l(m, n_2).$ By the universal property of tensor products, for every such fixed $l \in L,$ we have a unique $R$-module homomorphism $\phi_l : M \otimes N \to M \otimes ( N \otimes L)$ such that $\phi_l (m,n) = \varphi_l (m,n) = m \otimes (n \otimes l).$ Thus, the map $\psi : (M \otimes N) \times L \to M \otimes (N \otimes L)$ such that $\psi(m \otimes n, l) = \phi_l(m,n) = m \otimes (n \otimes l)$ (and then extended by linearity on the first argument) is well-defined. Moreover, $\psi$ is easily seen to be bilinear and thus (again by the universal property) induces a unique $R$-module homomorphism taking $(m \otimes n) \otimes l$ to $m \otimes (n \otimes l).$ Similarly, we can construct a homomorphism the other way around, inverse to this one, hence the isomorphism.

QED

We thus see that for a commutative ring $R,$ we can unambiguously construct the k-fold tensor product $M_1 \otimes M_2 \otimes \ldots \otimes M_k$ of k $R$-modules $M_1, \ldots M_k.$ Moreover, from the universal property of tensor product and the last remark in the last post, we deduce that this k-fold tensor product is the universal object with respect to k-multilinear maps :

Theorem (Universal property): Let $R$ be a commutative ring and let $M_1, \ldots M_k, L$ be $R$-modules. Let $\iota : M_1 \times \ldots \times M_k \to M_1 \otimes \ldots \otimes M_k$ be such that $\iota(m_1, \ldots, m_k) = (m_1 \otimes \ldots \otimes m_k).$ Let $\varphi : M_1 \times \ldots M_k \to L$ be a k-multilinear function. Then there exists a unique $\phi : M_1 \otimes \ldots \otimes M_k \to L$ such that $\varphi = \phi \circ \iota.$

Once again, we deduce the existence of a bijection between the set of k-multilinear functions from $M_1 \times \ldots M_k$ to $L$ and the set of $R$-module homomorphisms from $M_1 \otimes \ldots \otimes M_k$ to $L.$

Reference: D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.