We are looking at a function $F:\mathbb{N}^\mathbb{N}\to \mathbb{N}^\mathbb{N}$ $$F(h)=h\circ h$$ I want to show that $F$ is not surjective.
I have a solution, but I'm not quite sure if this is a proof or I just missed something. Let's take the function $h(x)=x$ and feed it to $F$. We will get $F(h(x))=h(h(x))$. The function $h(h(x))$ is always equal to $x$. So I say that $F$ is not surjective, because if I feed $h(x)$ in, some values like for instance $t(x)=x^2$ will not be hit in the codomain, so $F$ is not surjective.
I am really unsure whether this proves the statement. I feel like it's too easy. If it's wrong, I'd really appreciate some tipps!