Clarification of Proof, Theorem 2.2.4(b) Silverman The theorem states:
"Let $\iota:K(C_2) \to K(C_1)$ be an injection of function fields fixing $K$. Then there exists a unique nonconstant map $\phi:C_1 \to C_2$ defined over $K$ such that $\phi^* = \iota$."
Here, $\phi^*$ is the induced injection of function fields from a morphism $\phi$. The proof states the function $\phi = [1, \iota(X_1/X_0), \dots, \iota(X_N/X_0)]$ works, but I don't see why. Given an $f \in K(C_2)$, one would want to prove
\begin{equation}
\iota(f) = f(1, \iota(X_1/X_0), \dots \iota(X_N/X_0))
\end{equation}
which I don't quite see as to why this is true. Is there something perhaps about the behavior of $\iota$ that I'm missing?
 A: I'm posting this answer now only because, having had exactly the same question today, and having only understood why the claim is true thanks to a friend, I think a more detailed explanation may benefit future readers of AEC. (I should note, however, that I think much of what follows is what KReiser alludes to in their comment above, albeit phrased in a way that avoids appealing to schemes.)
Since $C_2$ is a projective variety, it contains at least one point, i.e., we have $P = [x_1, \dots, x_n] \in C_2$ with some $x_i \neq 0$. Without loss of generality, Silverman supposes $i = 0$. This means that $C_2$ is not contained the hyperplane $X_0 = 0$. The significance of this is that the affine piece of $C_2$ obtained by dehomogenizing with respect to $X_0$ is nonempty and so we work with this affine subvariety $V$. By definition (see chapter 1, page 10 in the second edition), we take $K(C_2)$ to be the function field of $V$. This function field is $K(g_1, \dots, g_n)$, where $g_i$ is the function on $K(C_2)$ corresponding to $X_i/X_0$.
It follows that $\iota(K(C_2)) = \iota(K(g_1, \dots, g_n)) = K(\iota(g_1), \dots, \iota(g_n))$. In particular, for $f \in I(V)$, $\iota(f(g_1(X), \dots, g_n(X))) = f(\iota(g_1(X)), \dots, \iota(g_n(X))) \in I(V')$, where $V'$ is the affine subvariety of $C_1$ obtained by dehomogenizing with respect to $X_0$. So we have a rational map from $\varphi : V \to V'$ defined by $\varphi = (\iota(g_1), \dots, \iota(g_n))$. Identifying $V$ and $V'$ with subsets of $C_1$ and $C_2$, respectively, via the standard inclusion maps into projective space yields a rational map $\phi : C_1 \to C_2$ with $\phi = [1, \iota(g_1), \dots, \iota(g_n)]$. This $\phi$ is precisely the map Silverman defines in the proof.
