The infinitesimal Torelli problem

Let $g\ge 2$ and let the Torelli map denote the map sending a genus $g$ curve to its principally polarized Jacobian. We verify the well known fact that the map induced on tangent spaces by the Torelli map is dual to the multiplication map $$\mathrm{S}\mathrm{y}\mathrm{m}$^2 H^0(C, \omega_C) \rightarrow H^0(C, \omega_C^{\otimes 2})$.