  Nominal sets provide a foundation for reasoning about names. They are used
primarily in syntax with binders, but also, e.g., to model automata over
infinite alphabets. In this paper, nominal sets are related to nominal renaming
sets, which involve arbitrary substitutions rather than permutations, through a
categorical adjunction. In particular, the left adjoint relates the separated
product of nominal sets to the Cartesian product of nominal renaming sets.
Based on these results, we define the new notion of separated nominal automata.
These automata can be exponentially smaller than classical nominal automata, if
the semantics is closed under substitutions.
