Let ${\cal C}$ and ${\cal D}$ be $A_{\infty}$-pre-categories over $k$. An $A_{\infty}$-functor $F:{\cal C}\to{\cal D}$ is a functor between the corresponding non-unital $A_{\infty}$-pre-categories such that $F$ takes quasi-isomorphisms in ${\cal C}$ to quasi-isomorphisms in ${\cal D}$.