TITLE: Understanding Hartshorne's proof that every projective morphism is proper.
QUESTION [5 upvotes]: In chapter $II$ of Hartshorne, theorem $4.9$ shows that every projective morphism is proper, using the valuative criterion for properness. I understand how the required morphism is constructed, but I don't understand why it is unique. I would greatly appreciate an explanation explaining why "The uniqueness of this morphism follows from the construction and the way the $V_i$ patch together."
I can see why any morphism with image lying inside $V_k$, the affine open piece $D_+(X_k)$, is unique but I don't see why it should be the case that the image of any morphism of the correct form should lie inside $V_k$.

REPLY [3 votes]: Just to suggest a different approach: the uniqueness is equivalent to the fact that a projective morphism is separated (the valuative criterion for separatedness), and this can be shown by directly showing the diagonal morphism $\mathbb P^n \to \mathbb P^n \times \mathbb P^n$ is a closed immersion.  (In fact, since formation of diagonals and closed immersions are both compatible with base-change, it suffices to check this over $\mathbb Z$.)   Now you could try to check directly that the diagonal is closed, say by computing how it interacts with the standard cover of $\mathbb P^n$ by $n+1$ affine spaces.