An $A_{\infty}$-category with strict identity morphisms is a non-unital $A_{\infty}$-category ${\cal C}$, such that for any object $X$ there exists an element $1=1_{X}\in Hom^{0}(X,X)$ (identity morphism) such that $m_{2}(1,f)=m_{2}(f,1)=f$ and $m_{n}(f_{1},...,1,...,f_{n})=0,n\neq 2$ for any morphisms $f,f_{1},...,f_{n}$.