We have to remind ourselves who the Jacobian $J(C)$ is. One quick and dirty answer is that the Jacobian of a curve $C$--seen as a riemann surface--is just the quotient $H_1(C, \mathbb{R}) /H_1(C, \mathbb{Z})$ -- this is an insufficient definition since I have not specified either a metric or a Kahler form. But it's sufficient to convince us that the induced map $J(\phi): J(C) \to J(C')$ ought to be $H_1(\phi)$ factored through the quotient. Again, the above description of $J(C)$ is incomplete but sufficient our purpose.

Any riemannian manifold has a meaningful associated Jacobi variety. In the case of complex curves, there's that alternative description as the quotient $H^0(C, \Omega^1)^* / jH_0(C, \mathbb{Z})$ where $j: C \to H_1(C, \mathbb{Z})$ is the Abel-Jacobi period mapping (defined up to a choice of base point on $C$--but all homotopy equivalent). Here we find that the induced map $J'(C)$ between jacobians is $\phi^*$ where one deserves to convince themselves that map actually factors through the quotient.