Addendum to “On Meager Additive and Null Additive Sets in the Cantor space $2^{\omega} $ and in $\mathbb R$” (Bull. Polish Acad. Sci. Math. 57 (2009), 91–99)

We prove in ZFC that there is a set $A\subseteq 2^{\omega}$ and a surjective function $H : A\to \langle0,1\rangle $ such that for every null additive set $X\subseteq \langle0,1)$, $H^{-1}(X)$ is null additive in $2^\omega$. This settles in the affirmative a question of T. Bartoszyński.