  Binary multirelations can model alternating nondeterminism, for instance, in
games or nondeterministically evolving systems interacting with an environment.
Such systems can show partial or total functional behaviour at both levels of
alternation, so that nondeterministic behaviour may occur only at one level or
both levels, or not at all. We study classes of inner and outer partial and
total functional multirelations in a multirelational language based on relation
algebra and power allegories. While it is known that general multirelations do
not form a category, we show that the classes of deterministic multirelations
mentioned form categories with respect to Peleg composition from concurrent
dynamic logic, and sometimes quantaloids. Some of these are isomorphic to the
category of binary relations. We also introduce determinisation maps that
approximate multirelations either by binary relations or by deterministic
multirelations. Such maps are useful for defining modal operators on
multirelations.
