Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object. The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and multiplication given by the monoid structure $(\mu,\eta)$.
I would like to see an explicit proof, if it exists, of the following statement:
The monad described above is commutative.
In particular, I'm looking for a reference on the topic.
Directly refering to the definition at the nlab, the morphism $\alpha : T(A) \otimes T(B) \to T(A \otimes B)$ is given by the composition $$\small (A \otimes M) \otimes (B \otimes M) \cong (A \otimes (B \otimes M)) \otimes M \cong ((A \otimes B) \otimes M) \otimes M \cong (A\otimes B) \otimes (M \otimes M) \to (A\otimes B) \otimes M,$$ which is $$(a \otimes m) \otimes (b \otimes m') \mapsto (a \otimes b) \otimes (m' \cdot m)$$in element notation (see Section 3.3 in my thesis what element notation roughly means). A similar reasoning shows that the morphism $\beta: T(A) \otimes T(B) \to T(A \otimes B)$ is given by $$(a \otimes m) \otimes (b \otimes m') \mapsto (a \otimes b) \otimes (m \cdot m').$$ Thus, equality is just the commutativity of the monoid $(M,\mu,\eta)$.
A reference for this is Example 6.3.12 in my thesis, but I am sure that A. Kock already remarked this in one of his papers on commutative monads.
