Transfers of $A_{\mathrm{\infty }}$- and other homotopy structures as Grothendieck bifibrations

We show that the functor which assigns to an A-infinity morphism between isotopy classes of A-infinity algebras whose linear part is a chain homotopy equivalence its underlying chain map is a discrete Grothendieck bifibration. We then generalize our results to P-infinity structures over a field of characteristic zero, for any quadratic Koszul operad P. An immediate application is a categorical framework in which the transfers of e.g. A-infinity, L-infinity and C-infinity structures are strictly functorial. A by product of our reasoning is a general transfer theorem for P-infinity algebras, which we prove in the last section.