nLab
n-functor

An $n$-functor is simply a functor between $n$-categories. Similarly, an $\mathrm{\infty }$-functor is a functor between $\mathrm{\infty }$-categories.

Of course, as the definition of $n$-category gets more complicated as $n$ increases, so does the appropriate definition of functor. This explains why one says ‘$n$-functor’ instead of simply ‘functor’ all along. On the other hand, anything that goes between $n$-categories, if it deserves to be called anything like ‘functor’ at all, will be an $n$-functor, so the prefix is not really necessary.

Special cases:

• function
• functor
• 2-functor, including pseudo functor