I understand that a combinatorial class $\mathcal{A}$ is a set of objects, with a function of size $|\cdot|_{\mathcal{A}}:\mathcal{A}\to \mathbb{N}$. With objects of size n: $\mathcal{A}_n=\{\alpha\in\mathcal{A}\;:\; |\alpha|_{\mathcal{A}}=n\}$, and a condition of finitude $\#\mathcal{A}_n<\infty$. 

In a book ["Introduction to Combinatorial Enummeration" by Dvid G. Wagner][1] chapter 11, I came across the term "class of structures", and they define it as: 

A class $\mathcal{A}$ associated with a finite set $X$ is another finite set $\mathcal{A}_X$, such that is $X$ and $Y$ are finite sets:

If $X\neq Y$ then $\mathcal{A}_X\cap \mathcal{A}_Y=\emptyset$.

If $X$ and $Y$ are finite sets with $|X|=|Y|$, then $|\mathcal{A}_X|=|\mathcal{A}_Y|$.

My question is whether these two definitions refer to the same thing. I am somewhat confused


  [1]: https://uwaterloo.ca/combinatorics-and-optimization/sites/ca.combinatorics-and-optimization/files/uploads/files/co330-notes.pdf