Quiver motivic Hall convolution is associative and unital

Let $\alpha ,\beta ,\gamma$ be homogeneous classes of degrees $\mathbf{d},\mathbf{e},\mathbf{f}$. The iterated fiber products defining $(\alpha \ast \beta )\ast \gamma$ classify a representation $E$ with a flag $0\subset F_1\subset F_2\subset E$ whose successive factors $F_1$, $F_2\mathrm{/}{F}_1$, and $E\mathrm{/}{F}_2$ have dimensions $\mathbf{f},\mathbf{e},\mathbf{d}$. The maps to the three inputs are the same classes on these three factors.

The iterated fiber products defining $\alpha \ast (\beta \ast \gamma )$ classify the same flags: first $F_2$ is the subrepresentation of $E$ with quotient of dimension $\mathbf{d}$, then $F_1$ is its subrepresentation with quotient of dimension $\mathbf{e}$. For representations of a quiver, kernels, quotients and inverse images are computed vertexwise and inherit the arrow maps, so the two descriptions are inverse equivalences of the flag stacks.

Both parenthesizations therefore pull back the same exterior product of $\alpha ,\beta ,\gamma$ to the same three-step flag stack and push it forward along the same map sending a flag to its middle representation. Functoriality of pullback, pushforward and exterior product in [F2] gives $(\alpha \ast \beta )\ast \gamma =\alpha \ast (\beta \ast \gamma )$. Bilinearity extends this identity from homogeneous classes to all of $H_Q$.

If one factor is the zero representation, the exact sequence with zero subobject or zero quotient is uniquely isomorphic to the identity sequence on the other factor, including its endpoint automorphisms. Thus the degree-zero class acts as a two-sided unit.