= Solution
Let $u:M'\to M$ be an <R-module homomorphism>. Naturality in the left argument means that precomposition by $u\otimes1_N$ on the left corresponds under the <Tensor-Hom adjunction> to precomposition by $u$ on the right. For $f:M\otimes_RN\to P$,
$$
\begin{aligned}
\Phi_{M',N,P}\bigl(f\circ(u\otimes1_N)\bigr)(m')(n)
&=f\bigl(u(m')\otimes n\bigr)\\
&=\Phi_{M,N,P}(f)\bigl(u(m')\bigr)(n).
\end{aligned}
$$
Thus the naturality square commutes pointwise on every $m'\in M'$ and $n\in N$.
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