Morphisms of pre-Calabi-Yau categories and morphisms of cyclic $A_{\infty}$-categories

In this article we prove that there exists a relation between $d$-pre-Calabi-Yau morphisms introduced by M. Kontsevich, A. Takeda and Y. Vlassopoulos and cyclic $A_{\infty}$-morphisms, extending a result proved by D. Fern\'andez and E. Herscovich. This leads to a functor between the category of $d$-pre-Calabi-Yau structures and the partial category of $A_{\infty}$-categories of the form $\mathcal{A}\oplus\mathcal{A}^*[d-1]$ with $\mathcal{A}$ a graded quiver and whose morphisms are the data of an $A_{\infty}$-structure on $\mathcal{A}\oplus\mathcal{B}^*[d-1]$ together with $A_{\infty}$-morphisms $\mathcal{A}[1]\oplus\mathcal{B}^*[d]\rightarrow \mathcal{A}[1]\oplus\mathcal{A}^*[d]$ and $\mathcal{A}[1]\oplus\mathcal{B}^*[d]\rightarrow \mathcal{B}[1]\oplus\mathcal{B}^*[d]$.