This appendix provides an explicit description of the free n-category generated by an n-polygraph. This section is mostly inspired of the work of Makkai. A formal definition of the syntax of n-categories is first provided, describing morphisms in an (n+1)-category freely generated by an n-polygraph, allowing reasoning by induction on its terms to prove results on free categories. It turns out that this syntax for n-categories, which corresponds to the one used throughout the book, is very "redundant", in the sense that there are many ways to express a composite of cells which will give rise to the same result, and is sometimes not very practical for this reason. An alternative syntax, which suffers less from these problems, is provided by restricting compositions. Finally, a brief mention of the word problem for free n-categories is made.