  Categories, $n$-categories, double categories, and multicategories (among
others) all have similar definitions as collections of cells with composition
operations. We give an explicit description of the information required to
define any higher category structure which arises as algebras for a familially
representable monad on a presheaf category, then use this to describe various
examples relating to higher category theory and cubical sets. The proof of this
characterization avoids tedious naturality arguments by passing through the
theory of categorical polynomials.
