How to call a function a function where the repetition of a function is redundant

Question

I have a very simple question, yet very hard to Google, so I thought I would ask you guys.

Given a function $f\colon V\rightarrow V$, and given that $f(\vec{v}) = f(f(\vec{v}))$.

What's this property called? I believe it's something from category theory. And what (simple) properties can one derive from it?

Answer 1

This property is called idempotency.

An idempotent linear map is also called a projection.

If $\Phi:V \to V$ is idempotent, then

$\ker\Phi \oplus \operatorname{im} \Phi = V$

The only possible eigenvalues of $\Phi$ are $0$ and $1$.