  Quantum supermaps provide a framework in which higher order quantum processes
can act on lower order quantum processes. In doing so, they enable the
definition and analysis of new quantum protocols and causal structures.
Recently, key features of quantum supermaps were captured through a general
categorical framework, which led to a framework of higher order process
theories (HOPT). The HOPT framework models lower and higher order
transformations in a single unified theory, with its mathematical structure
shown to coincide with the notion of a closed symmetric monoidal category. Here
we provide an equivalent construction of the HOPT framework from four simple
axioms of process-theoretic nature. We then use the HOPT framework to establish
connections between foundational features such as causality, determinism and
signalling, alongside exploring their interaction with the mathematical
structure of *-autonomy.
