In this MSE post, the author gives translations of the definition of morphism in Hartshorne, into more concrete terms. I have been able to verify that these are equivalent to the definition in Hartshorne, but I'm left with a nagging doubt. I think that the map $\phi : \mathbb{P}^1 \to \mathbb{P}^1$ given by $\phi = [X^2 + Y^2 : XY]$ should be a morphism, because it is everywhere defined and given explicitly by polynomials. From the equivalences linked however, I think I should be able to check this by looking at the induced map on each affine piece. On on the piece where $Y = 1$, the map is given by $x \mapsto \frac{x^2 + 1}{x}$, where $x = \frac{X}{Y}$. This map is not regular, however, because the regular functions on $\mathbb{A}^1$ are just the polynomial functions.
So is the map $\phi$ a morphism? And if so, where is my reasoning for verifying this going wrong?
When you restrict to $D(y)$, you don't get a morphism $D(y) \rightarrow D(y)$, you get a morphism $D(y) \rightarrow \mathbb{P}^1_k$.
The morphism sends $(x, 1) \in D(y)$ to $(x^2 + 1, x)$, which is equal to $ (\frac{x^2+1}{x}, 1)$ if $x \neq 0$, and $(1, 0)$ if $x = 0$. This is a well-defined morphism $D(y) \rightarrow \mathbb{P}^1_k$