ct.category theory – Why does each functor define an invariant, but is not every invariant always functorial? Examples?

In category theory, each functor defines an invariant, but not all invariants are functional

Why ?

Can you give some examples when

  • a functor is an invariant
  • an invariant is a functorial
  • an invariant is not functorial

I also need another example:

– a functor is not an invariant $ => $ it's absurd and false but I need you to show me the underlying contradiction

Idea

For $ A $ a monoid equipped with an action on an object $ V $, a invariant of action is a generalized element of $ V $ which is taken by the action to itself, hence a fixed point for all operations in the monoid.

A action of a category $ C $ on a tray $ S $ nothing is more than one functor $ rho: C to $ Together.