  Following on from the notion of (first-order) causality, which generalises
the notion of being tracepreserving from CP-maps to abstract processes, we give
a characterization for the most general kind of map which sends causal
processes to causal processes. These new, second-order causal processes enable
us to treat the input processes as 'local laboratories' whose causal ordering
needs not be fixed in advance. Using this characterization, we give a
fully-diagrammatic proof of a non-trivial theorem: namely that being
causality-preserving on separable processes implies being 'completely'
causality preserving. That is, causality is preserved even when the 'local
laboratories' are allowed to have ancilla systems. An immediate consequence is
that preserving causality is separable processes is equivalence to preserving
causality for strongly non-signalling (a.k.a. localizable) processes.
