I mean Endofunctor maps category $A$ to category $A$ - so all of it's objects, and morphisms must remain the same. Isn't this an identity Functor?
For example, if I have a category $A$ of integers with an ordering on them:
$O = \{1,2,3,4\}$
$M(x,y) = \{(x,y) | x,y \in O, x \leq y)\}$
any Endofunctor I try to think of ends up being an identity Functor...
What I am thinking wrong here? Thanks.
EDIT: As per suggestion, let me try mapping all objects, via and endofunctor $F$ to $1$ and all morphisms to $1 \leq 1$.
$F_{O}: O_{A} \to O_{A}, F_M: M_{A} \to M_{A}$
then $F_{O}(1) = 1, F(2) = 1, ...$ and $F_{M}( M(1,2) ) = M(1,1) $
so now either:
- Nothing is changed and $A$ remains the same (endofunctor having only "mapped" things around)
- Or the new $A$ is now missing all objects except for $1$ and all morphisms except for $1\leq 1$.
If it's the former (1), what is the point of this functor in the first place, if it does not "do" anything...