Does the Coend have a right adjoint? Let $ C $ be a small category and $ \mathfrak{X} $ be complete and cocomplete category. The coend may be seen as a functor
$ \int ^C : Fun (C\times C^{op}, \mathfrak{X} )\to \mathfrak{X} $
Indeed, $ \int ^C = colim\circ T $, in which $ T $ is a functor which associates each object of $ Fun (C\times C^{op} , \mathfrak{X}) $ to the appropriate coequalizer diagram.
So, I wonder if this functor $T$ has a right adjoint. I tried to find such a right adjoint, but I couldn't. Unfortunately, I couldn't use the Freyd adjoint theorem: despite the fact that $ \int ^C $ preserves colimits.
 A: The coend is a special case of the notion of a weighted colimit. Indeed, let $H : \mathcal{C}^\mathrm{op} \times \mathcal{C} \to [(\mathcal{C}^\mathrm{op} \times \mathcal{C})^\mathrm{op}, \mathbf{Set}]$ be the Yoneda embedding and let $W = \int^\mathcal{C} H$. It can be shown (using the Yoneda lemma for ends/coends) that $W (c, c') \cong \mathcal{C} (c', c)$, and for any $\mathcal{X}$ where coends exist, there is the following natural bijection:
$$\mathcal{X} (\int^\mathcal{C} F, T) \cong [(\mathcal{C}^\mathrm{op} \times \mathcal{C})^\mathrm{op}, \mathbf{Set}](W, \mathcal{X}(F, T))$$
Thus, $\int^\mathcal{C} F$ is the weighted colimit $W \star F$. Thus, it suffices to show that $W \star {-}$ has a right adjoint for a general weight $W$.
Let $\mathcal{J}$ be any small category and let $W : \mathcal{J}^\mathrm{op} \to \mathbf{Set}$ be a weight. Suppose $\mathcal{X}$ is complete and cocomplete. Then, for any diagram $F : \mathcal{J} \to \mathcal{X}$,
\begin{align}
\mathcal{X} (W \star F, T) 
& \cong [\mathcal{J}^\mathrm{op}, \mathbf{Set}] (W, \mathcal{X} (F, T)) \\
& \cong \int_{j : \mathcal{J}} \mathbf{Set} (W j, \mathcal{X} (F j, T)) \\
& \cong \int_{j : \mathcal{J}} \mathcal{X} (F j, W j \pitchfork T) \\
& \cong [\mathcal{J}, \mathcal{X}] (F, W \pitchfork T)
\end{align}
where $W j \pitchfork T$ denotes the $W j$-fold cartesian product of $T$. Thus,
$$W \star {-} \dashv W \pitchfork {-} : \mathcal{X} \to [\mathcal{J}, \mathcal{X}]$$
In the case where $W$ is the terminal weight, this recovers a well-known adjunction:
$$\varinjlim_\mathcal{J} \dashv \Delta : \mathcal{X} \to [\mathcal{J}, \mathcal{X}]$$
