Let $C$ be a compact Riemann surface, let $C^2$ be the cartesian square of $C$, let $J(C)$ be the degree zero Jacobian of $C$, and let $\delta : C^2 \to J(C)$ be the map $(x,y) \mapsto [\mathcal{O}(x-y)]$.

In this paper http://arxiv.org/abs/math/9810054 of Hain and Reed, page 9, they say that it is an elementary exercise in algebraic topology to show that $\delta^\ast(\phi) = \Delta + (\psi_1 + \psi_2)/2$.

Explanation of notation:

$\phi$ denotes the symplectic form $\sum dx_i \wedge dy_i$ on $J(C)$, where $x_i, y_i$ are coordinates on the torus $J(C) = H^1(C;\mathcal{O}) / H^1(C;\mathbb{Z})$ corresponding to a symplectic basis of $H_1(C;\mathbb{Z}) \cong H^1(C;\mathbb{Z})$. This is also the class of the theta divisor $\Theta$.

$\psi_i$ is the first Chern class of the relative cotangent bundle of the $i$th projection $C^{2} \to C$.

$\Delta$ is the class of the diagonal $C \to C^{2}$, $x \mapsto (x,x)$.

My question is: How to do this "elementary exercise"? It ought to be easy but I'm just not seeing it...