Let $C_1\subset \mathbb{P}^{N_1}$ and $C_2\subset \mathbb{P}^{N_2}$ be two curves. Then a map $\phi:C_1\to C_2$ can be defined as
$\phi=[f_0,\ldots,f_{N_2}],$
where each $f_i$ is a homogeneous polynomial. Such a map induces a map $\phi^*:K(C_2)\to K(C_1)$ given by $g\mapsto g\circ \phi$. What confuses me here is that the rings $K(C_i)$ are defined by Silverman to be the rings $K(C_i\cap \mathbb{A}^{N_i})$, so e.g. $K(C_2)$ is of the form $K[X_1,\ldots,X_{N_2}]/I$ and I have trouble seeing what $g\circ \phi$ should be for some $g\in K[X_1,\ldots,X_{N_2}]/I$, since $\phi$ has sort of "one component too much".
Is the idea here that we should essentially write $\phi$ in the form
$\phi = [f_0/f_i,\ldots,1,\ldots,f_{N_2}/f_i]$
where $i$ corresponds to our chosen embedding $\mathbb{A}^{N_2}\hookrightarrow U_i=\{X_i\neq 0\}\subset \mathbb{P}^{N_2}$? Now given some $g(X_1,\ldots,X_{N_2})\in K(C_2)$, we get
$\phi^*g=g\circ \phi =?$
How is this map supposed to be concretely represented? I know Hartshorne uses the representation where $K(C_2)$ is just quotients of homogeneous polynomials of the same degree, so the composition is pretty obvious. I'm trying to understand how to work with Silverman's definition.
EDIT: I think the idea here is to think of $K(C_2)$ as the ring generated by $X_0/X_i,\ldots,X_{N_2}/X_i$ and $K(C_1)$ as the ring generated by $Y_0/Y_j,\ldots,Y_{N_1}/Y_j$, so the polynomial $g$ is actually $g(X_0/X_i,\ldots,X_{N_2}/X_i)$. Then the map would be given by
$\phi^*g = g\left(\frac{f_0(Y_0/Y_j,\ldots,Y_{N_1}/Y_j)}{f_i(Y_0/Y_j,\ldots,Y_{N_1}/Y_j)},\ldots,\frac{f_{N_1}(Y_0/Y_j,\ldots,Y_{N_1}/Y_j)}{f_i(Y_0/Y_j,\ldots,Y_{N_1}/Y_j)}\right).$
Can anyone confirm this? This does seem to coincide with Hartshorne's definition, though checking it would be quite a mess...