  We provide a new characterisation of quantum supermaps in terms of an axiom
that refers only to sequential and parallel composition. Consequently, we
generalize quantum supermaps to arbitrary monoidal categories and operational
probabilistic theories. We do so by providing a simple definition of
locally-applicable transformation on a monoidal category. The definition can be
rephrased in the language of category theory using the principle of naturality,
and can be given an intuitive diagrammatic representation in terms of which all
proofs are presented. In our main technical contribution, we use this
diagrammatic representation to show that locally-applicable transformations on
quantum channels are in one-to-one correspondence with deterministic quantum
supermaps. This alternative characterization of quantum supermaps is proven to
work for more general multiple-input supermaps such as the quantum switch and
on arbitrary normal convex spaces of quantum channels such as those defined by
satisfaction of signaling constraints.
