  We exhibit an isomorphism of associative algebras between the
$\operatorname{Ext}$-algebra $\operatorname{Ext}_\Lambda^\ast(\Delta,\Delta)$
of standard modules over the dual extension algebra $\Lambda$ of two directed
algebras $B$ and $A$ and the dual extension algebra of the
$\operatorname{Ext}$-algebra $\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$
with $A$. There are natural $A_\infty$-structures on these
$\operatorname{Ext}$-algebras, and, under certain technical assumptions on $B$,
we describe that on $\operatorname{Ext}_\Lambda^\ast(\Delta,\Delta)$ completely
in terms of that on $\operatorname{Ext}_B^\ast(\mathbb{L},\mathbb{L})$. As an
example, we compute these $A_\infty$-structures explicitly in the case where
$B=A=K\mathbb{A}_n /(\operatorname{rad}K\mathbb{A}_n)^\ell$.
