An $A_{\infty}$-functor $F:{\cal C}\to{\cal D}$ between $A_{\infty}$-pre-categories is called an $A_{\infty}$-equivalence functor if:

a) Every object $Y\in{\cal D}$ is quasi-isomorphic to an object $\phi(X),X\in{\cal C}$.

b) The functor induces quasi-isomorphisms of non-unital $A_{\infty}$-algebras of morphisms, corresponding to all transversal sequences of objects.