Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Denote $(\_)^*:=\text{Hom}_R(\_,R)$. There is a natural map \begin{align} \Phi_M: M \otimes_R M^* & \longrightarrow \text{Hom}_R(M^*,M^*) \\ x \otimes y & \longmapsto (f\mapsto f(x)y) \end{align}
My question is:
If there exists an isomorphism $M\otimes_RM^*\cong \text{Hom}_R(M^*,M^*)$, then is it true that $\Phi_M$ is an isomorphism?
My reason for asking this question:
It is well known that if $M$ is a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$ such that $M\cong M^{**}$, then the natural evaluation map \begin{align} M & \longrightarrow M^{**} \\ x & \longmapsto (f \mapsto f(x)) \end{align} is an isomorphism. The map defined in my question is kind of similar, so I wonder if that is true as well.
My thoughts:
If we have isomorphic finitely generated modules $X \cong Y$ and a linear surjection $f:X\to Y$, then $f$ is an isomorphism. Indeed, let $g:Y\to X$ be an isomorphism. Then, $g\circ f: X\to X$ is a surjection, hence it is an isomorphism (by Nakayama), so $f$ is injective, and we are done. Hence, under the hypothesis of my question, it would be enough to show $\Phi_M$ is surjective...
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