it is well known that "fukaya category" is in fact an $a_{\infty}$-pre-category in sense of kontsevich and soibelman. the reason is that in general the morphism spaces are defined only for transversal pairs of lagrangians, and higher products are defined only for transversal sequences of lagrangians. it has been conjectured by kontsevich and soibelman that for any graded commutative ring $k,$ quasi-equivalence classes of $a_{\infty}$-pre-categories over $k$ are in bijection with quasi-equivalence classes of $a_{\infty}$-categories over $k$ with strict (or weak) identity morphisms. in this talk i will sketch a proof of this conjecture for essentially small $a_{\infty}$-(pre-)categories, in the case when $k$ is a field. in particular, it follows that we can replace fukaya $a_{\infty}$-pre-category with a quasi-equivalent actual $a_{\infty}$-category.