What is a Nontrivial Functor in Category Theory?

In category theory, a functor is called "nontrivial" or "nontenable" if it is not an equivalence relation. In other words, if the functor does not preserve the equality of morphisms, then it is nontrivial.

For example, consider the category of sets, where the morphisms are functions between sets. The identity functor, which simply maps each set to itself and each function to itself, is a trivial functor because it preserves all morphisms. On the other hand, the functor that maps each set to its powerset and each function to its inverse is nontrivial because it does not preserve the equality of morphisms.

In general, a nontrivial functor can be thought of as a "non-trivial" transformation between categories, which changes the underlying structure of the category in some way.