TITLE: Does every category have a functor?
QUESTION [5 upvotes]: Is there any one (or more) categories that doesn't have a functor? Functors go between categories, so is there any category that only has an identity functor but no other functor that maps it to another category?

REPLY [2 votes]: Every category has at least 2 outgoing functors: the identity functor plus at least another one. 
Actually every category has infinitely many outgoing functors.
You can easily see this by considering the cases of:

the empty category which goes to every other one (it is the initial category, after all)
the non empty categories which go trivially or non trivially to every other non empty and/or non terminal category

However the empty category has only one ingoing functor. The identity functor. Every other category has more than one ingoing functor