Free functor from the category of G-nominal sets to G-01-nominal sets

The category Nom, of all finitely supported G-sets, called the G-nominal sets, where G = Permf (D) is the group of finitary permutations over a countable infinite set D, is a subject of interest by both set theorists and computer scientists. The category 01-Nom, of all G-nominal sets equipped with the source and the target operations, was introduced by Pitts. In this paper, we study the existence of the free objects in the category 01-Nom over G-nominal sets.