An $A_{\infty}$-pre-category is a non-unital $A_{\infty}$-pre-category ${\cal C}$, satisfying the following extension property:

For any finite collection of transversal sequences $S_{1},...,S_{m}$ in ${\cal C}$ and an object $X$ there exist objects $X_{+}$ and $X_{-}$ and quasi-isomorphisms $f_{-}:X_{-}\to X$, $f_{+}:X\to X_{+}$ such that extended sequences $(X_{-},S_{1},...,S_{m},X_{+}),1\leq i\leq m$ are transversal.