For every commutative ring $A$, let $\mathbf{Symm}_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}_{\mathbb{Z}}$.

We can define a bilinear map $\boxdot : \mathbf{Symm}_{\mathbb{Q}} \times \mathbf{Symm}_{\mathbb{Q}} \to \mathbf{Symm}_{\mathbb{Q}}$ by setting

$p_{\lambda} \boxdot p_{\mu} = \prod\limits_{i\geq 1,\ j\geq 1} p_{\operatorname*{lcm}\left(\lambda_i,\mu_j\right)}^{\gcd\left(\lambda_i,\mu_j\right)}$

for any two partitions $\lambda = \left(\lambda_1,\lambda_2,\lambda_3,...\right)$ and $\mu = \left(\mu_1,\mu_2,\mu_3,...\right)$. Here, we are writing $\boxdot$ as an infix operator (that is, $a\boxdot b$ means $\boxdot\left(a,b\right)$), and $p_\nu$ means the $\nu$-power sum symmetric function.

This bilinear map $\boxdot$ is associative. I call it the "arithmetic product", as it boils down to the arithmetic product of species viewed through the cycle index series.

Now, species theory can be used to show that $\boxdot$ restricts to a well-defined map $ \mathbf{Symm} \times \mathbf{Symm} \to \mathbf{Symm}$ (that is, the restriction of $\boxdot$ to $\mathbf{Symm} \times \mathbf{Symm}$ has its image in $\mathbf{Symm}$). My question is: Can this be proven more elementarily? Is there a good way to describe this map on an actual basis of $ \mathbf{Symm}$ rather than on the power-sum symmetric functions? Is there a more direct combinatorial or even representation-theoretical significance of this map?

(This is somewhat similar to MO question #120924, where another operation on $\mathbf{Symm}$ is constructed on the power sums first and then happens to be integral for weird reasons.)