  The task of constructing compositional semantics for network-style
diagrammatic languages, such as electrical circuits or chemical reaction
networks, has been dubbed the black boxing problem, as it gives semantics that
describes the properties of each network that can be observed externally, by
composition, while discarding the internal structure. One way to solve these
problems is to formalise the diagrams and their semantics using hypergraph
categories, with semantic interpretation a hypergraph functor, called the black
box functor, between them. Reviewing a principled method for constructing
hypergraph categories and functors, known as decorated corelations, in this
paper we construct a category of \emph{decorating data}, and show that the
decorated corelations method is itself functorial, with a universal property
characterised by a left Kan extension. We then argue that the category of
decorating data is a good setting in which to construct any hypergraph functor,
giving a new construction of Baez and Pollard's black box functor for reaction
networks as an example.
