  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
$\mathcal{A}[1]\oplus\mathcal{B}^*[d]\rightarrow
\mathcal{B}[1]\oplus\mathcal{B}^*[d]$.
