Given three quivers $(X_0, X_1, \sigma, \tau)$, $(Y_0, Y_1, \phi, \psi)$, and $(Z_0, Z_1, \chi, \omega)$, and two morphisms $F := (F_0 : X_0 \to Y_0, F_1 : X_1 \to Y_1) : (X_0, X_1, \sigma, \tau) \to (Y_0, Y_1, \phi, \psi)$ and $G := (G_0 : Y_0 \to Z_0, G_1 : Y_1 \to Z_1) : (Y_0, Y_1, \phi, \psi) \to (Z_0, Z_1, \chi, \omega)$ of quivers, what is the condition for $F$ and $G$ to be composable, i.e., what is minimally required to let $G \circ F : (X_0, X_1, \sigma, \tau) \to (Z_0, Z_1, \chi, \omega)$ be a morphism of quivers again?
Of course requiring that $\phi = \omega$ would be sufficient, but $(G_1 \circ \omega) \circ F_0 = (G_1 \circ \phi) \circ F_0$ would do as well. But now I wonder if it can be relaxed further, or even worse: that I'm allowing too many morphisms to be composable than should be.