This order is not accidental - functors start making sence in context of natural transformations and categories are rarely interesting without reasoning in terms of mappings between them i.e. functors.
Did you know that mathematicians used the term natural transformation before it was ever defined? Even before categories and functors - things that are suppose to show up earlier in chronological order. For people involved in abstract algebra those days things seemed natural without giving any definition of naturality - especially when they observe commutative diagrams. Actually you can perceive naturality as a synonymous of the obvious choice or the only one.
Work in progress...