  Suppose that $\alpha$ is an action of the semigroup $\mathbb{N}^{2}$ on a
$C^*$-algebra $A$ by endomorphisms. Let $A\times_{\alpha}^{\textrm{piso}}
\mathbb{N}^{2}$ be the associated partial-isometric crossed product. By
applying an earlier result which embeds this semigroup crossed product (as a
full corner) in a crossed product by the group $\mathbb{Z}^{2}$, a composition
series $0\leq L_{1}\leq L_{2}\leq A\times_{\alpha}^{\textrm{piso}}
\mathbb{N}^{2}$ of essential ideals is obtained for which we identify the
subquotients with familiar algebras.
