  We specialise a recently introduced notion of generalised dinaturality for
functors $T : (\mathcal{C}^\text{op})^p \times \mathcal{C}^q \to \mathcal{D}$
to the case where the domain (resp., codomain) is constant, obtaining notions
of ends (resp., coends) of higher arity, dubbed herein $(p,q)$-ends (resp.,
$(p,q)$-coends). While higher arity co/ends are particular instances of
"totally symmetrised" (ordinary) co/ends, they serve an important technical
role in the study of a number of new categorical phenomena, which may be
broadly classified as two new variants of category theory.
  The first of these, weighted category theory, consists of the study of
weighted variants of the classical notions and construction found in ordinary
category theory, besides that of a limit. This leads to a host of varied and
rich notions, such as weighted Kan extensions, weighted adjunctions, and
weighted ends.
  The second, diagonal category theory, proceeds in a different (albeit
related) direction, in which one replaces universality with respect to natural
transformations with universality with respect to dinatural transformations,
mimicking the passage from limits to ends. In doing so, one again encounters a
number of new interesting notions, among which one similarly finds diagonal Kan
extensions, diagonal adjunctions, and diagonal ends.
