Let $F:k\text{-Sch} \rightarrow \text{Set}$ be the following functor:
$$F(X)=\text{Hom}_{X-\text{sch}}(\mathbb A^1_X, \mathbb A^1_X).$$
I would like to show that such functor is not representable.
I am able to show that is not representable by an affine scheme. Indeed, suppose $X=\text{Spec }R$ represents such functor and let $p(x)\in \text{Hom}_{X-\text{sch}}(\mathbb A^1_X, \mathbb A^1_X)=R[x]$ the universal object. If $A$ is a $k$-algebra and $a(x)\in A[x]$, then it should exist a morphism of $k$-algebras $R\rightarrow A$, such that the induced morphism $R[x]\rightarrow A[x]$ maps $p(x)$ to $a(x)$. If the degree of $p(x)$ is smaller than the one of $a(x)$, then it is impossible. Such argument could be extented to prove that such functor it is not representable by a quasi-compact scheme, but I'm not able to do the general case.
Any hints or ideas?