  In this article we study the homotopy theory of pre-Calabi-Yau morphisms,
viewing them as Maurer-Cartan elements of an $L_{\infty}$-algebra. We give two
different notions of homotopy: a notion of weak homotopy for morphisms between
$d$-pre-Calabi-Yau categories whose underlying graded quivers on the domain
(resp. codomain) are the same, and a notion of homotopy for morphisms between
fixed pre-Calabi-Yau categories $(\mathcal{A},s_{d+1}M_{\mathcal{A}})$ and
$(\mathcal{B},s_{d+1}M_{\mathcal{B}})$. Then, we show that the notion of
homotopy is stable under composition and that homotopy equivalences are
quasi-isomorphisms. Finally, we prove that the functor constructed by the
author in a previous article between the category of pre-Calabi-Yau categories
and the partial category of $A_{\infty}$-categories of the form
$\mathcal{A}\oplus\mathcal{A}^*[d-1]$, for $\mathcal{A}$ a graded quiver,
together with hat morphisms sends homotopic $d$-pre-Calabi-Yau morphisms to
weak homotopic $A_{\infty}$-morphisms.
