Hom weak ω-categories of a weak ω-category

Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak -categories, take more sophisticated approaches, and the nature of the 'hom is often not immediate from the definitions'. In this paper, we focus on Leinster's definition of weak -category based on an earlier definition by Batanin and construct, for each weak -category, an underlying (weak -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak -category. We also show that our construction is functorial with respect to weak -functors introduced by Garner.