Lemma 4.1 says：let $X$ and $Y$ be two varieties, and let $\phi$ and $\psi$ be two morphisms from $X$ to $Y$, and suppose there is a nonempty open subset $U \subseteq X$ such that $\phi|_U = \psi|_U$, then $\phi = \psi$.
The proof assumes that $Y \subseteq \mathbb{P}^n$ for some $n$. But every projective variety can be covered by affine varieties(Corollary 2.3, P11). And proposition 4.3 says: every variety $Y$ has a base for the topology consisting of open affine subsets.
My question is why we do not assume $Y \subseteq \mathbb{A}^n$ be an affine variety? What's the difference between Corollary 2.3 and proposition 4.3？(In proposition 4.3 assume $Y$ is projective)
Thank you vwry much.
The statement that $Y$ is covered by affine varieties does not mean $Y \subset \mathbb{A}^n$ for some $n$. For example, this will never be true for $Y = \mathbb{P}^1$. On the other hand, every variety (by Hartshorne's Chapter I definition) sits inside of $\mathbb{P}^n$ for some $n$.