n-category = (n,n)-category
n-groupoid = (n,0)-category
Of course, as the definition of -category gets more complicated as increases, so does the appropriate definition of functor. This explains why one says ‘-functor’ instead of simply ‘functor’ all along. On the other hand, anything that goes between -categories, if it deserves to be called anything like ‘functor’ at all, will be an -functor, so the prefix is not really necessary.