The Green Hall coproduct is coassociative

Fix basis objects $A,B,C,R$. The coefficient of $[A]\otimes [B]\otimes [C]$ in $({\mathrm{\Delta }}^{\mathrm{\prime }}\otimes \mathrm{i}\mathrm{d}){\mathrm{\Delta }}^{\mathrm{\prime }}([R])$ is $\frac{{\nu }^{⟨A+B,C⟩+⟨A,B⟩}}{a_R}{\sum }_{[L]}\frac{P_{L,C}^R{P}_{A,B}^L}{a_L}\mathrm{.}$ Only finitely many $L$ occur: each is a middle term of an extension of $A$ by $B$, and ${\mathrm{YExt}}^1(A,B)$ is finite.

Using [F2], each summand in the flag count becomes $a_A{a}_B{a}_C{g}_{L,C}^R{g}_{A,B}^L$. It counts a flag $C\subseteq U\subseteq R$ with $U\mathrm{/}C\cong B$ and $R\mathrm{/}U\cong A$, weighted by the same endpoint automorphism factor $a_A{a}_B{a}_C$.

The coefficient of $[A]\otimes [B]\otimes [C]$ in $(\mathrm{i}\mathrm{d}\otimes {\mathrm{\Delta }}^{\mathrm{\prime }}){\mathrm{\Delta }}^{\mathrm{\prime }}([R])$ is $\frac{{\nu }^{⟨A,B+C⟩+⟨B,C⟩}}{a_R}{\sum }_{[T]}\frac{P_{A,T}^R{P}_{B,C}^T}{a_T}\mathrm{.}$ Again the sum is finite by finiteness of ${\mathrm{YExt}}^1(B,C)$. Using [F2], it counts flags $C\subseteq T\subseteq R$ with $T\mathrm{/}C\cong B$ and $R\mathrm{/}T\cong A$, with the same weight $a_A{a}_B{a}_C$. The two flag sets described in step 2.1 and in this calculation are identical under $U=T$.

By bilinearity [F3], $⟨A+B,C⟩+⟨A,B⟩=⟨A,B+C⟩+⟨B,C⟩$. Hence the two coefficients are equal. Every triple coefficient is therefore equal, proving coassociativity in the coefficientwise completion.

In ${\mathrm{\Delta }}^{\mathrm{\prime }}([R])$, the coefficient of $[R]\otimes [0]$ and that of $[0]\otimes [R]$ are both one: the exact sequences with zero endpoint are uniquely determined up to the $a_R$ endpoint labellings, which cancel the denominator $a_R$. Applying $ϵ$ to either factor therefore returns $[R]$, and all other terms are killed. Linearity proves the two counit identities.