How does a correspondence on an algebraic curve $C$ induce a map on $\Omega^1_C$? Apparently it passes through the Jacobian of $C$ but I don't quite understand it.

More specifically, I was reading a paper that said roughly the following:

Let C be a curve and $\Gamma$ (with maps $\pi_i: \Gamma \rightarrow C$ for $i=1,2$) be a correspondence on $C$. The map $\pi_2$ is a double cover and $\tau: \Gamma \rightarrow \Gamma$ is a map that switches the elements of the fibers of $\pi_2$. Then the induced map on $H^0(C, \Omega^1)$ is given by: $\omega \mapsto \pi_1^* \omega + \tau^* \pi_1^* \omega$, where the differentials of $C$ are identified with the ones of $\Gamma$ that are $\tau^*$-invariant.

I don't understand why the map on $H^0(C, \Omega^1)$ is what it is claimed to be (even assuming the mentioned identification). I guess this follows from general theory of correspondences but I don't know where I would find such a statement.