Finite-quiver Hall multiplication counts extensions by automorphism orbits

Fix the middle object to be $M$. The coefficient of ${\delta }_M$ in ${\delta }_U\star {\delta }_V$ counts subobjects $S\subseteq M$ with $S\cong U$ and $M\mathrm{/}S\cong V$, by [F1] and [F2]. For each such $S$, the monomorphisms $U\to M$ with image $S$ form a set of size $\mathrm{\mid }\mathrm{Aut}(U)\mathrm{\mid }$; the epimorphisms $M\to V$ with kernel $S$ form a set of size $\mathrm{\mid }\mathrm{Aut}(V)\mathrm{\mid }$. These choices are independent. Hence the number of endpoint-fixed short exact sequences with middle object literally $M$ is $g_{V,U}^M\mathrm{\mid }\mathrm{Aut}(U)\mathrm{\mid }\mathrm{\mid }\mathrm{Aut}(V)\mathrm{\mid },$ where $g_{V,U}^M$ is the Hall coefficient of [F1].

The finite group $\mathrm{Aut}(M)$ acts on those sequences by $(i,p)\mapsto (gi,p{g}^{-1})$. Its orbits are exactly the endpoint-fixed Yoneda classes with middle term isomorphic to $M$: an isomorphism from any middle object $E\cong M$ transports its sequence to one on $M$, and changing that isomorphism changes the result by this action. The stabilizer of a sequence $(i,p)$ consists of $g$ with $gi=i$ and $pg=p$. Put $h=g-1$. Since $hi=0$ and $p$ is an epimorphism, $h$ factors uniquely as $h=ifp$ for a unique $f:V\to U$. Conversely, $1+ifp$ is invertible because $(ifp{)}^2=0$, with inverse $1-ifp$. Thus every stabilizer has exactly $\mathrm{\mid }\mathrm{Hom}(V,U)\mathrm{\mid }$ elements.

For a fixed sequence, the map from $\mathrm{Aut}(M)$ to its orbit has fibers exactly the cosets of its stabilizer, so that orbit has size $\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{Hom}(V,U)\mathrm{\mid }$. Summing over the $\mathrm{\mid }{\mathrm{YExt}}^1(V,U{)}_M\mathrm{\mid }$ orbits gives $g_{V,U}^M\mathrm{\mid }\mathrm{Aut}(U)\mathrm{\mid }\mathrm{\mid }\mathrm{Aut}(V)\mathrm{\mid }=\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }\frac{\mathrm{\mid }{\mathrm{YExt}}^1(V,U{)}_M\mathrm{\mid }}{\mathrm{\mid }\mathrm{Hom}(V,U)\mathrm{\mid }}\mathrm{.}$ Multiplying the coefficient of ${\delta }_M$ in ${\kappa }_U\star {\kappa }_V$ by the definition of ${\kappa }_M$ yields precisely the claimed coefficient. Summing over the finitely many possible middle terms is valid by [F3] and [F4].