 Extension complex of quiver representations

Put $C^0={⨁}_i\mathrm{Hom}(X_i,Y_i)$ and $C^1={⨁}_{\rho :i\to j}\mathrm{Hom}(X_i,Y_j)$. The differential has components $g_{\rho }{h}_i-h_j{f}_{\rho }$. Its kernel is ${\mathrm{Hom}}_Q(X,Y)$ and its cokernel is ${\mathrm{Ext}}_Q^1(X,Y)$. In a vertexwise splitting of an extension, the off-diagonal arrow blocks change by these coboundaries when the splitting changes.