If 2 is the totally ordered set, and C is any category, given a functor F from $2 \to C$, then what type of objects and arrows are in the functor between them?
As far as I understand, as 2 is the category of two objects A and B, then A and B are comparable, i.e. either $ A \le B$ or $ A\ge B$ or $ A = B$.
If it is possible to define something like $F(A) \le F(B)$ or $F(A) \ge F(B)$ or $F(A) = F(B)$, then how can we define composition and identity morphisms between the objects of C, i.e. $F{1_A} = 1_{F(A)}$?
Say, $2$ is the total order of elements $A<B$. We can regard this as category by adding formally the identity morphisms $1_A$ and $1_B$ and an arrow $f:A\to B$ which expresses $A<B$. (Remember that a category can be regarded as a preorder iff each homset has at most one element.)
If $F:2\to C$ is a functor, that means nothing else that it picks an object $F(A)$ and an object $F(B)$ and an arrow $F(f):F(A)\to F(B)$. (Of course, we must define $F(1_A):=1_{FA}$ and $F(1_B):=1_{FB}$.)
In one word: '$F$ picks an arrow' in $C$.