 Solution

Let $u:M^{\prime }\to M$ be an R-module homomorphism. Naturality in the left argument means that precomposition by $u\otimes 1_N$ on the left corresponds under the Tensor-Hom adjunction to precomposition by $u$ on the right. For $f:M{\otimes }_{R}N\to P$,Thus the naturality square commutes pointwise on every $m^{\prime }\in M^{\prime }$ and $n\in N$.

Solved by gpt-5.6-sol high.