Pullbacks and pushouts Say you have a curve $X$ of genus $g \geq 2$, and a surjective map $\phi:X \to E$, where $E$ is an elliptic curve. Denote by $J_X$, $J_E$ the Jacobians of $X$, $E$ respectively. Then we get an induced pushout map ${\phi}_\ast : J_X \to J_E$ and an induced pullback map $\phi^\ast: J_E \to J_X$. My questions are: what do these maps do, explicitly? Also, what are $\phi^\ast \phi_{\ast}$ and $\phi_{\ast} \phi^\ast$? 
As an aside, is there a good reference where I can learn more about pushout and pullback maps in more generality (with a view towards algebraic geometry and concrete uses)?
Thanks!
 A: A partial answer: You can learn more about pullbacks & pushouts in a much more abstract setting with Category Theory.
A: First of all $J_X =  \text{Div}_0(X) / \text{P}(X)$, the divisors of $X$ of degree 0 modulo the principle divisors of $X$.
So $\phi \colon X \to E$ induces a map $\phi_\# \colon \text{Div}_0(X) \to \text{Div}_0(E)$ given by $\phi_\#(\sum n_i P_i) = \sum n_i \phi(P_i)$. This $\phi_\#$ maps principle divisors to principle divisors, so in turn induces the map $\phi_* \colon J_X \to J_E$.
For the other map, note that the preimage $\phi^{-1}(Q)$ of a point $Q$ of $E$ consists of finitely many points, the number of which does not depend on $Q$ (provided that you count the points in the preimage with the appropriate multiplicity). So $\phi^{-1}(Q)$ can be seen as a divisor of $X$ and so $\phi$ induces a map $\phi^\# \colon \text{Div}(E) \to \text{Div}(X)$ given by $\phi^\#(\sum_i n_i Q_i) = \sum_i n_i \phi^{-1}(Q_i)$. Divisors of degree 0 map to divisiors of degree 0 under this map, so this $\phi^\#$ restricts to a map $\phi^\# \colon \text{Div}_0(E) \to \text{Div}_0(X)$. This map $\phi^\#$ also maps principle divisors to principle divisors, so induces the map $\phi^* \colon J_E \to J_X$.
