Cyclic $A_\infty$-algebras and double Poisson algebras

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 M. Kontsevich. We also show correspondence is functorial in quite satisfactory way, giving rise to (partial) functor from the category of dg partial algebras. Finally, further generalize it include double $P_{\infty}$-algebras, as introduced T. Schedler.