Let $R$ be a commutative ring with $1\_R$. Let $M$ be a $R$-module and $F$ be a free $R$-module.

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> How do I find a specific isomorphism to show that $$\mathrm{Hom}(M, R)\otimes\_R F \cong \mathrm{Hom}(M, F).$$
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Due to the universal property I know that there exists a unique homomorphism $$\varphi: \mathrm{Hom}(M, R)\otimes\_R F \rightarrow \mathrm{Hom}(M, F),$$
because $\beta: \mathrm{Hom}(M, R) \times F \rightarrow \mathrm{Hom}(M, F)$ with $\beta: (\lambda, w) \mapsto \lambda(v)\cdot w$ is bilinear.

How do I go about finding $\varphi$? Can I use $\beta = \varphi \circ \tau$, where $\tau$ is the natural inclusion or do I need to use the resemblance to dual spaces. How much does the case with $R$ a field, and $M, F$ vectorspaces differ from the one with rings?