On a strange homotopy category

For an additive category $\mathcal C$ in which each morphism has a kernel, it is proved that the homotopy category of the category of complexes over $\mathcal C$ which are concentrated in degrees 2,1,0 and are exact in degrees 2 and 1 is abelian. Under assumption that a category $\mathcal C$ is abelian, earlier this result was obtained by considering the heart of a suitable $t$-structure on the homotopy category of $\mathcal C$.