Let ${\cal C}$ be a non-unital $A_{\infty}$-category. Let us replace spaces of morphisms by their cohomology with respect to $m_{1}$. In other words, we define $Hom_{H({\cal C})}(X,Y):=\{Ker\,m_{1}\}/\{Im\,m_{1}\}$, where $m_{1}:Hom_{\cal C}(X,Y)\to Hom_{\cal C}(X,Y)[1]$ is the composition map. Then $H({\cal C})=({\cal C},Hom_{H({\cal C})}(\cdot,\cdot))$ gives rise to a “non-unital” category structure with the class of objects ${\cal C}$ and composition of morphisms induced by $m_{2}$. We write “non-unital” because there are no identity morphisms $id_{X}\in Hom_{H({\cal C})}(X,X)$.