Defining multiplication transformation for free monoid monad on monoidal category I'm learning about monads in Riehl's Category Theory in Context, and after reading an example about the free monoid monad (also known as the list monad to computer scientists) on the monoidal category $(\mathsf{Set}, \times, *)$, I've reached an exercise meant to demonstrate that this monad can in fact be defined on any monoidal category with
"coproducts that distribute over the monoidal product." To summarize the example, the list monad is the functor with action $T(A) = \coprod_{n \geq 0} A^n$ induced by the free $\dashv$ forgetful adjunction between sets and monoids, where the unit components are the coproduct inclusions and the multiplication components are concatenations. (e.g. $((a,b),(c)) \mapsto (a, b, c)$.)
In the exercise, the reader is given a generic monoidal category $(V, \otimes, *)$ with finite coproducts which $\otimes$ "preserves in each variable." (A footnote clarifies this with the equality $(v \sqcup v') \otimes (w \sqcup w') = v \otimes w \sqcup v' \otimes w \sqcup v \otimes w' \sqcup v' \otimes w'$. Isn't this "the monoidal product distributing over the coproduct," the reverse of the author's claim?) Then the reader is asked to define the unit and multiplication ($\eta : 1_V \Rightarrow T$ and $\mu : T^2 \Rightarrow T$) so that $T(X) = \coprod_{n \geq 0} X^{\otimes n}$ is a monad. (Tangent: shouldn't $V$ be required to have countable rather than just finite coproducts for this to make sense?) In analogy with the example, it's clear to me that the the unit components should again be the coproduct inclusions, but I'm having a really hard time defining the multiplication.
If I understand correctly, in the example the singleton $*$ is both the unit object and the terminal object. That would allow me to take the morphism $T(A)^{k-1} \to *$ and apply $T(A) \times (-)$ to get $T(A)^k \to T(A) \times * \cong T(A)$ for any $k$, defining a cone from the diagram $\emptyset, T(A), T(A)^2, \dots$ to $T(A)$ and thus inducing a unique morphism $T^2(A) \to T(A)$ by the universal property which I can take to be $\mu_A$. But not only am I unable to verify that this coincides with the example's concatenation, this also doesn't seem to be a useful analogy for the exercise since, as far as I can tell, there's no guarantee that $V$ has a terminal object at all!
EDIT: To clarify my response to Kevin Arlin's comment, the coproduct inclusions $A^k \xrightarrow{\iota} T(A)$ are split monomorphisms in $\mathsf{Set}$, (any injective function restricts to a bijection onto its image) giving us morphisms $T(A) \xrightarrow{r} A^k$ for any $k$ such that $r \circ \iota = \mathrm{id}_{A^k}$. This allows us to form diagrams as in the following example:

Here, $r \circ (a) = a$ and $r \circ (b) = b$, so it should be that $\tilde{r} \circ ((a), (b)) = (a,b)$, which is exactly the desired concatenation for pairs of singlet lists. Similar constructions define concatenations of triples, quadruples, etc. of lists into $T(A)$, and hopefully induce a unique map $T^2(A) \to T(A)$. But now my concerns become:

*

*This construction is specific to pairs of singlet lists. Each different combination of list lengths requires a unique such morphism. Can this collection of morphisms in $\mathsf{Set}(T(A)^2, T(A))$ define a single morphism?

*Are coproduct inclusions in $V$ still split monic?

 A: Based on Kevin Arlin's final comment, I was quickly able to construct the components $\mu_X$ of the multiplication to my satisfaction. Unfortunately, in the weeks since then I've still been unable to show that $\mu$ is natural in $X$ or that $\mu$ and $\eta$ satisfy the monad commutation relations. (Aside from $\mu_X\eta_{T(X)} = \mathrm{id}_{T(X)}$, which follows easily from the construction of $\mu$.) I've grown tired and frustrated with this problem, however, and will be moving on. But for the sake of posterity, I'll post my construction.
As Kevin mentioned, distributivity of the monoidal product yields $T(X)^2 \cong \coprod_{i,j} X^i \otimes X^j$. (Here I'm abusing notation and writing $X^{\otimes i}$ as $X^i$ for clarity, though I understand that this is distinct from the categorical product.) More generally, we can say that $T(X)^N \cong \coprod_{a \in \mathbb{N}^N} \bigotimes_{i=1}^N X^{a_i}$. And since $\bigotimes_{i=1}^N X^{a_i} \cong X^{\sum_{i=1}^N a_i}$ by associativity, this defines a cocone under the discrete diagram of all such monoidal powers of $X$ with nadir $T(X)$, and hence induces a morphism $T(X)^N \xrightarrow{\alpha_N} T(X)$.

As $N$ varies, these morphisms $\alpha_N$ further define a cocone under the discrete diagram of all monoidal powers of $T(X)$ with nadir $T(X)$. The multiplication component $\mu_X$ is the unique morphism induced by the universal property.

Since $\alpha_1$ is clearly the identity morphism on $T(X)$, this proves the sole commutation relation that I have been able to show:
$$
\mathrm{id}_{T(X)} = \alpha_1 = \mu_X\iota_{T(X)} = \mu_X\eta_{T(X)}.
$$
I welcome anyone who is able to show naturality of $\mu$ in $X$ and the remaining monad commutation relations, but I no longer wish to spend my energy thinking about this problem.
