How do I build $n$-categories and $n$-functors, starting from a $0$-category and a $0$-functor?
We can take examples, focusing on $n=2$. And we state clearly what kinds of $m$-morphisms are allowed.
Here are some basics we can start with:
A 0-category is a set.
A 1-category is an ordinary category.
A 2-category is (depending on how strict was your initial notion of $\infty$-category) either a strict 2-category or a bicategory.
A $n$-functor is a functor between $n$-categories.