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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.
Wilson et al. (Mon,) studied this question.