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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. $F{1_A} = 1_{F(A)}$?

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    $\begingroup$ Instead of guessing, you should work with the definitions. They will immediately answer the question. The task is not to forget about the definitions, but rather to become used to them. $\endgroup$
    – Martin Brandenburg
    Jan 8, 2014 at 19:16

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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 $C$.

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