  In this article, we extend results of J. Leray and B. Vallette on homotopical
properties of pre-Calabi-Yau algebras to the case of pre-Calabi-Yau categories.
We give direct proofs of the results adapting techniques used by D. Petersen
for the case of coalgebras, instead of using properadic calculus. More
precisely, we prove that given quasi-isomorphic dg quivers and a pre-Calabi-Yau
structure on one of them there exists a pre-Calabi-Yau structure on the other
as well as a pre-Calabi-Yau morphism between the two. Moreover, we show that a
pre-Calabi-Yau morphism whose first component is an isomorphism of dg quivers
is an isomorphism in the category of pre-Calabi-Yau categories. We incidentally
show that any pre-Calabi-Yau category has a minimal model. Finally, we prove
that any quasi-isomorphism of pre-Calabi-Yau categories admits a quasi-inverse.
