Consider the setting of categories enriched over a suitable monoidal category $\mathbb V$.
We define $$\mathrm{Dist}(X,Y):=\mathbb V−\mathrm{Cat}(X^ \mathrm{op}⊗Y,\mathbb V).$$
Recall the definition of ends. Taking the end is an operation of signature
$$\mathrm{Dist}(X,X)→\mathbb V.$$
QUESTION: Is there an analogue for functors into V of higher arity. More explicitly: Is there a canonical operation of signature
$$\mathbb V−\mathrm{Cat}(X^?⊗X^?⊗X^?,V)→\mathbb V$$
where the ? are to be replaced by either op or nothing.
MOTIVATION: I like to think of the composition $\otimes$ of (2-ary)distributors and the right adjoints to $D\otimes−$ and $−\otimes E$ as "horn-filling" (in the sense of viewing categories as simplicial sets).
I hope to find a similar situation for "3-ary distributors" - Whenever there is a (oriented?) tetrahedron of 3-ary functors with one side missing we should be able to find the missing side.
So given
$$U\in\mathrm{Dist}^3(X,Y,A)$$ $$V\in\mathrm{Dist}^3(Y,Z,A)$$ $$W\in\mathrm{Dist}^3(Z,X,A)$$
there should be $$?(U,V,W)\in\mathrm{Dist}^3(X,Y,Z)$$
and related adjoints.
Of course $\mathrm{Dist}^3$ remains to be defined.
