Quiver representation morphism ($h_{t(\rho )}{f}_{\rho }=g_{\rho }{h}_{s(\rho )}$)

For quiver representations $X=(X_i,f_{\rho })$ and $Y=(Y_i,g_{\rho })$, a morphism consists of maps $h_i:X_i\to Y_i$ satisfying $h_{t(\rho )}{f}_{\rho }=g_{\rho }{h}_{s(\rho )}$ for every arrow. Composition is vertexwise. These families are the kernel of the extension complex of quiver representations.

A quiver representation morphism is an isomorphism exactly when all its vertex maps are invertible. Choosing bases turns this into the base change action on quiver representations, whose orbits coincide with isomorphism classes.