Associativity of tensor products for compatible bimodules

For fixed $p\in P$, the pairing $(m,n)\mapsto m\otimes (n\otimes p)$ is $R$-balanced: $(mr,n)$ and $(m,rn)$ have the same image by the outer left $R$-action in [L1]. Thus [L2] and [L3] give a homomorphism $a_p:M{\otimes }_{R}N\to M{\otimes }_R(N{\otimes }_{S}P)$.

For fixed $m\in M$, the pairing $(n,p)\mapsto (m\otimes n)\otimes p$ is $S$-balanced, so [L2] and [L3] give $b_m:N{\otimes }_{S}P\to (M{\otimes }_{R}N){\otimes }_{S}P$.

The pairing $(x,p)\mapsto a_p(x)$ from $(M{\otimes }_{R}N)\times P$ is $S$-balanced. On generators, $a_p((m\otimes n)s)=m\otimes ((ns)\otimes p)=m\otimes (n\otimes sp)=a_{sp}(m\otimes n)$ by [L1], and additivity extends the equality to every $x$. Hence [L2] gives ${\alpha }_{M,N,P}$ with the displayed formula.

The pairing $(m,y)\mapsto b_m(y)$ is $R$-balanced. It is enough to check generators $y=n\otimes p$, where $b_{mr}(n\otimes p)=((mr)\otimes n)\otimes p=(m\otimes rn)\otimes p=b_m((rn)\otimes p)=b_m(r(n\otimes p))$ by [L1]. Therefore [L2] gives $\beta :M{\otimes }_R(N{\otimes }_{S}P)\to (M{\otimes }_{R}N){\otimes }_{S}P$.

The composite $\beta \alpha$ fixes every tensor $(m\otimes n)\otimes p$, and $\alpha \beta$ fixes every tensor $m\otimes (n\otimes p)$; successive applications of uniqueness in [L2] show that these composites are the identity maps.

The same elementary-tensor calculation shows compatibility with any outer actions, and replacing $m,n,p$ by their images under compatible homomorphisms proves naturality because both candidate composites agree on all elementary tensors.

Thus ${\alpha }_{M,N,P}$ is the asserted canonical natural isomorphism with inverse $\beta$.