I'm studying Gathmann's notes (https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2021/alggeom-2021.pdf), and I'm currently stuck in Exercise 5.7:
Show:
(a) Every morphism $f:\mathbb{A}^1\setminus\{0\}\to \mathbb{P}^1$ can be extended to a morphism $\mathbb{A}^1\to \mathbb{P}^1$.
(b) Not every morphism $f:\mathbb{A}^2\setminus\{0\}\to \mathbb{P}^1$ can be extended to a morphism $\mathbb{A}^2\to \mathbb{P}^1$.
I know that questions have been asked here (see Extending morphisms between varieties, Extending a morphism $\mathbb{A}^1-0\to \mathbb{P}^1$ to $\mathbb{A}^1$). But none of them uses a construction of $\mathbb{P}^1$ as in Gathmann: In the notes, $\mathbb{P}^1$ is constructed by gluing together two copies of $\mathbb{A}^1$ along $\mathbb{A}^1\setminus\{0\}$ using the gluing isomorphism $x\mapsto \frac1x$.
My question is, how should I understand morphisms to $\mathbb{P}^1$? I know that they are continuous maps such that the pullback of a regular function is regular, but the definition of the structure sheaf on $\mathbb{P}^1$ is quite complicated here: it is defined by $$\mathcal{O}_{\mathbb{P}^1}(U)=\{\phi:U\to K:i_1^\ast\phi\in\mathcal{O}_{X_1}(i_1^{-1}(U))~\text{and}~i_2^\ast\phi\in\mathcal{O}_{X_2}(i_2^{-1}(U))\}$$ where $i_1,i_2:\mathbb{A}^1\to \mathbb{P}^1$ are two natural embeddings.
More precisely, in (b), I tried to construct a morphism $f:\mathbb{A}^2\setminus\{0\}\to\mathbb{P}^1$ via $$f(x,y)=\begin{cases}[x/y] &\text{if}~y\neq 0 \\ [y/x] &\text{if}~x\neq 0\end{cases}$$ where $[\cdot]$ is the equivalence class of $\cdot$ in $\mathbb{P}^1$. I think this is well-defined, but I can't even show this is a morphism. Would anyone be able to help and enlighten me?
(I know that homogeneous coordinates can be used here, but using this definition of $\mathbb{P}^1$ it seems to be unsuitable.)