  Precategories generalize both the notions of strict $n$-category and
sesquicategory: their definition is essentially the same as the one of strict
$n$-categories, excepting that we do not require the various interchange laws
to hold. Those have been proposed as a framework in which one can express
semi-strict definitions of weak higher categories: in dimension 3, Gray
categories are an instance of them and have been shown to be equivalent to
tricategories, and definitions of semi-strict tetracategories have been
proposed, and used as the basis of proof assistants such as Globular. In this
article, we are mostly interested in free precategories. Those can be presented
by generators and relations, using an appropriate variation on the notion of
polygraph (aka computad), and earlier works have shown that the theory of
rewriting can be generalized to this setting, enjoying most of the fundamental
constructions and properties which can be found in the traditional theory,
contrarily to polygraphs for strict categories. We further study here why this
is the case, by providing several results which show that precategories and
their associated polygraphs bear properties which ensure that we have a good
syntax for those. In particular, we show that the category of polygraphs for
precategories form a presheaf category.
