A module-valued coend computed as a quotient of a direct sum

The displayed data define a functor into $\mathbb{Z}$-modules: the only non-identity square in ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ with nontrivial source factors through the slot $(1,0)$, and its codomain is $(0,1)$, where every structure map is the zero homomorphism, so the square commutes automatically.

By [L1] the coend is the direct sum of the two diagonal values modulo the submodule generated by the dinaturality differences coming from the off-diagonal value $T(1,0)=\mathbb{Z}$. Here the direct sum is $\mathbb{Z}\oplus \mathbb{Z}$, and for $n\in \mathbb{Z}$ the generator contributed by $u$ is ${\mathrm{ȷ}}_0(T(u,1_0)(n))-{\mathrm{ȷ}}_1(T(1_1,u)(n))=(2n,-3n)$; the identity morphisms contribute only $(0,0)$. So the coend is $Q:=(\mathbb{Z}\oplus \mathbb{Z})\mathrm{/}⟨(2,-3)⟩$.

The homomorphism $q:\mathbb{Z}\oplus \mathbb{Z}\to \mathbb{Z}$, $(a,b)\mapsto 3a+2b$, kills $(2,-3)$ and so descends, by [F2], to a homomorphism $\bar{q}:Q\to \mathbb{Z}$. If $3a+2b=0$, then $3a=-2b$, so $3$ divides $b$ and $2$ divides $a$; writing $a=2k$ and $b=-3k$ shows $\ker q=⟨(2,-3)⟩$. Hence $\bar{q}$ is an isomorphism, and under it the two quotient cowedge components are multiplication by $3$ and by $2$.

On the nontrivial test module $\mathbb{Z}$, a cowedge ${\lambda }_0,{\lambda }_1:\mathbb{Z}\to \mathbb{Z}$ is determined by the integers $a:={\lambda }_0(1)$ and $b:={\lambda }_1(1)$, and the cowedge equation at $u$ is $2a=3b$. So $a=3t$ and $b=2t$ for a unique $t\in \mathbb{Z}$, and multiplication by $t$ on $Q\cong \mathbb{Z}$ is the unique homomorphism factoring the cowedge through the displayed quotient. This is exactly the initial-cowedge property on that test module.