ag.algebraic geometry - Kodaira-Spencer Theory and moduli of curves

I was looking at a paper of Farkas and the following confusing point came up.

Let $mathscr{M}_g$ be the moduli stack of smooth genus $g$ curves and let $pi: mathscr{C} to mathscr{M}_g$ be the universal curve. Let $mathscr{F}$ be $Omega^1_pi otimes Omega^1_pi$, where $Omega^1_pi$ is the sheaf of relative differentials of $pi$. Then the pushforward $pi_* mathscr{F}$ is isomorphic $Omega^1_{mathscr{M}_g}$.

Why is this true? Farkas says this follows from Kodaira-Spencer theory. I googled for a while and asked a few students, but couldn't figure this out.