Fubini: an end over a product index category and the two iterated ends exist together and agree

Under the reindexing, $T$ is a functor of four slots, contravariant in the first two and covariant in the last two, and holding the two $\mathcal{C}$-slots fixed at a pair $(c_1,c_2)$ leaves a functor of the two $\mathcal{D}$-slots whose chosen end is $S(c_1,c_2)$, with counit ${\theta }_d^{(c_1,c_2)}$ natural in the parameter $(c_1,c_2)$ by [L2]. Symmetrically for $S^{\mathrm{\prime }}$ with the roles of $\mathcal{C}$ and $\mathcal{D}$ exchanged.

Fix an object $X$ of $\mathcal{E}$. Given a wedge $\omega$ from $X$ to $T$, [L1] makes it dinatural in each variable separately; dinaturality in $d$ at fixed $c$ is exactly the wedge equation for the family $({\omega }_{(c,d)}{)}_d$ over the integrand with the $\mathcal{C}$-slots held at $(c,c)$, so by [F1] there is exactly one ${\psi }_c:X\to S(c,c)$ with ${\theta }_d^{(c,c)}{\psi }_c={\omega }_{(c,d)}$ for every $d$. By [L3], applied with parameter category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$, that family $\psi$ is a wedge over $S$ if and only if every ${\theta }_d^{(c,c)}{\psi }_c$, which is ${\omega }_{(c,d)}$, is dinatural in $c$ at fixed $d$ — which is the other half of [L1]. Conversely a wedge $\psi$ over $S$ produces ${\omega }_{(c,d)}:={\theta }_d^{(c,c)}{\psi }_c$, dinatural in $d$ because ${\theta }^{(c,c)}$ is a wedge precomposed with ${\psi }_c$, and dinatural in $c$ by [L3] again, hence a wedge over $T$ by [L1].

The two assignments of step 2.1 are mutually inverse, since each $\psi$ is the unique factorisation of the $\omega$ it produces, and $\omega$ is recovered from $\psi$ by the defining equation. A morphism $h:X\to X^{\mathrm{\prime }}$ satisfies ${\omega }_{(c,d)}^{\mathrm{\prime }}h={\omega }_{(c,d)}$ for every $(c,d)$ exactly when it satisfies ${\psi }_c^{\mathrm{\prime }}h={\psi }_c$ for every $c$: one direction is composition with ${\theta }_d^{(c,c)}$, and the other is the uniqueness in [F1]. So the wedge category of $T$ and the wedge category of $S$ are isomorphic over the identity on vertices, terminal objects correspond, and ${\int }_{(c,d)}T(c,d,c,d)$ exists exactly when ${\int }_{c}S(c,c)$ does, with the same vertex.

The same argument with the roles of $\mathcal{C}$ and $\mathcal{D}$ exchanged, applied to the chosen family $S^{\mathrm{\prime }}$, gives that ${\int }_{(c,d)}T(c,d,c,d)$ exists exactly when ${\int }_d{S}^{\mathrm{\prime }}(d,d)$ does, again with the same vertex. Hence any one of the three objects exists exactly when the others do, and by [L4] any two choices of them are joined by exactly one isomorphism commuting with every component.

For the coend clause, let $U(x_1,x_2):=T(x_2,x_1)$ on objects and $U(g,h):=T(h,g)$ on morphisms, read as a functor $(\mathcal{C}\times \mathcal{D}{)}^{\mathrm{o}\mathrm{p}}\times (\mathcal{C}\times \mathcal{D})\to {\mathcal{E}}^{\mathrm{o}\mathrm{p}}$; it satisfies the functor laws by [F8] and [F6], since reversing both the pair of slots and the direction of the target twice returns the composition order of $T$. Its diagonal values are those of $T$, and by [F2] a wedge from $X$ to $U$ in ${\mathcal{E}}^{\mathrm{o}\mathrm{p}}$ is precisely a cowedge from $T$ to $X$ in $\mathcal{E}$, a morphism of wedges being a morphism of cowedges reversed; so a terminal wedge over $U$ is an initial cowedge under $T$, that is a coend of $T$. Applying steps 3.1 and 4.1 to $U$ in ${\mathcal{E}}^{\mathrm{o}\mathrm{p}}$, with the chosen family of inner coends of $T$ as the chosen family of inner ends of $U$, gives the coend clause in full.