A functor $F:{\cal C}\to{\cal D}$ between non-unital $A_{\infty}$-pre-categories is given by the following data:

1) A map of classes of objects $\phi:{\cal C}\to{\cal D}$, such that $\phi^{n}({\cal C}_{n}^{tr})\subset{\cal D}_{n}^{tr}$.

2) For any transversal sequence of objects $(X_{0},...,X_{n}),\,n\geq 1$ in ${\cal C}$, a morphism of graded $k$-modules

These data satisfy the following property: the sequence $f_{n},n\geq 1$ defines an $A_{\infty}$-morphism $\oplus_{i<j}Hom_{{\cal C}}(X_{i},X_{j})\to\oplus_{i<j}Hom_{{\cal D}}(\phi(X_{i}),\phi(X_{j}))$.