  In this article we prove that there exists an explicit bijection between nice
$d$-pre-Calabi-Yau algebras and $d$-double Poisson differential graded
algebras, where $d \in \mathbb{Z}$, extending a result proved by N. Iyudu and
M. Kontsevich. We also show that this correspondence is functorial in a quite
satisfactory way, giving rise to a (partial) functor from the category of
$d$-double Poisson dg algebras to the partial category of $d$-pre-Calabi-Yau
algebras. Finally, we further generalize it to include double
$P_{\infty}$-algebras, as introduced by T. Schedler.
