End and coend

Summary

An end is a universal way of extracting from a functor $S \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathcal{D}$ an object that is compatible with all the diagonal values $S(c, c)$. A coend is the dual notion. Ends are special limits and coends are special colimits. The set of natural transformations between two functors is an end, and many constructions such as tensor products and Kan extensions are coends or ends.