# Combinatorial Reasoning For Identity of Species

Given a combinatorial species $\mathcal{X},$ let $\mathcal{X}^{\bullet}$ be the pointing species that distinguishes a certain element (e.g. if $\mathcal{X}$ were the species of trees then $\mathcal{X}^{\bullet}$ would be the species of rooted trees).

Given two combinatorial species $\mathcal{X}$ and $\mathcal{Y}$ let $\mathcal{X} \times \mathcal{Y}$ denote the standard labeled product of the given species. We further use $\mathcal{X} \ast \mathcal{Y}$ to denote the (asymmetric) box product of $\mathcal{X}$ and $\mathcal{Y}$. This box product is the same as the labeled product, except that when labels are being distributed over $\mathcal{X}$ and $\mathcal{Y}$ the "smallest" label is always placed on the $\mathcal{X}$-structure.

Let $\mathcal{A}$, $\mathcal{B},$ and $\mathcal{C}$ be some combinatorial species. Is there a simple (non-algebraic way) to see that the conditions $$\mathcal{A} = \mathcal{B}\ast \mathcal{C}$$ and $$\mathcal{A}^\bullet = \mathcal{B}^\bullet \times \mathcal{C}$$ are equivalent?