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Let $R$ be a commutative ring. Show that $(R$-$Mod, \otimes ,R)$ is a symmetric monoidal category.

Furthermore, show that it is a closed symmetric monoidal category.

I've seen statements of these facts, but can't seem to find the proofs anywhere.

Here is the definition of a symmertric monoidal category https://ncatlab.org/nlab/show/symmetric+monoidal+category

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    The proofs are quite long. Are there any particular bits you're stuck on? For example would it be enough to state the definitions of the various coherence maps, leaving it for you to convince yourself that the relevant diagrams commute? – 2017-02-17
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    Once you know what $\otimes ,\alpha, \rho, \lambda$ are, you just slog through checking that the pentagon and triangle diagrams commute. The category is symmetric because any two R-modules $A$ and $B$ satisfy $A\otimes B\cong B\otimes A$ and this iso is its own inverse. – 2017-02-17
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    And how do we know it is closed? – 2017-02-17
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    It's closed because the tensor is left adjoint to the module of module morphisms. – 2017-02-17

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