etale sheafification of a presheaf

Question

Consider the following presheaf on the big etale site of smooth schemes over a field $k$: to every smooth $k$-scheme $U$, associate $$F(U):= \{f: U \to \mathbb A^1_k ~|~ \text{$f$ factors through the inclusion of a proper open subset of $\mathbb A^1_k$} \}.$$ What is the etale sheafification of this presheaf? Is it the sheaf represented by $\mathbb A^1$? Why?

Answer 1

Let $f : U \to \mathbb{A}^1$ be a morphism. Let $\{S_i \to \mathbb{A}^1\}$ be the collection of all proper open subsets of $\mathbb{A}^1$. This is an étale (even Zariski) cover. We pull them back along $f$ to obtain an étale cover $\{U_i \to U\}$. By construction each $f|_{U_i}$ belongs to $F(U_i)$. This shows that $F^+= \hom(-,\mathbb{A}^1)$.

There is nothing special about $\mathbb{A}^1$ here, it could be any scheme which is covered by proper open subsets.