The end and the coend of a functor ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$

Let $T:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}\to \mathcal{D}$ be a functor and let $\mathrm{Wd}(T)$ and $\mathrm{Cwd}(T)$ be the categories of wedges over $T$ and of cowedges under $T$ (Wedges and cowedges, and the categories they form).

An end of $T$ is a terminal object of $\mathrm{Wd}(T)$, and a coend of $T$ is an initial object of $\mathrm{Cwd}(T)$ (Initial object, terminal object, and zero object). In short: an end is a terminal wedge and a coend an initial cowedge. Neither need exist.

Written out, an end is a pair $(e,\omega )$ in which ${\omega }_c:e\to T(c,c)$ is a wedge and, for every wedge $(x,\xi )$ over $T$, there is exactly one morphism $u:x\to e$ with

${\omega }_c\circ u={\xi }_c\text{for every object }c\text{ of }\mathcal{C}\mathrm{.}$

A coend is a pair $(q,\rho )$ in which ${\rho }_c:T(c,c)\to q$ is a cowedge and, for every cowedge $(x,\xi )$ under $T$, there is exactly one morphism $v:q\to x$ with $v\circ {\rho }_c={\xi }_c$ for every $c$.

The vertex of an end is written ${\int }_{c}T(c,c)$ and the vertex of a coend ${\int }^{c}T(c,c)$, the subscripted integral denoting the end and the superscripted one the coend. The components ${\omega }_c$ of the terminal wedge are the projections of the end and the components ${\rho }_c$ of the initial cowedge the injections of the coend. The variable $c$ in the integral notation is bound: it names the dinatural variable and nothing else.

Because an end is a dinatural transformation (Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$) with a universal property rather than an element-level construction, the definition applies to any target category $\mathcal{D}$ whatever, and it never asserts existence. Which functors have ends, and in which targets, is a separate question answered by the comparison theorems on this page and by the hypotheses they carry.

The projections ${\omega }_c$ are indexed by the objects of $\mathcal{C}$, while the wedge equation they satisfy is indexed by its morphisms. Thus an end records the family of diagonal values together with every tie imposed through the off-diagonal values $T(c,c^{\mathrm{\prime }})$. For a discrete category there are no nonidentity ties and the end reduces to the product of the diagonal values.