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?
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
A very strong no! For every two categories $C,D$ and an object $d\in ob(D)$ there is a functor $F:C\to D$ mapping every thing in $C$ to $d$ and $id_d$.
For every category $C$ there is a functor from $C$ to the category $1$ consisting of exactly one object and one arrow.
So you can map every category $C$ to the category $1$.
Conversely, $1$ can be embedded into any other category.
So every category "has a functor" other than identity functor.