A functor $F:{\cal C}_{1}\to{\cal C}_{2}$ between non-unital $A_{\infty}$-categories is given by the following data:

1) A map of classes of objects $\phi:{\cal C}_{1}\to{\cal C}_{2}$.

2) For any finite sequence of objects $X_{0},...,X_{n}$, $n\geq 0$, a morphism of graded $k$-modules $f_{n}:\otimes_{0\leq i\leq n-1}Hom_{{\cal C}_{1}}(X_{i},X_{i+1})\to Hom_{{\cal C}_{2}}(\phi(X_{0}),\phi(X_{n}))[1-n].$

The following condition holds for any $X_{1},...,X_{N}\in{\cal C}_{1}$: the sequence $f_{n},n\geq 1$ defines an $A_{\infty}$-morphism