Let $f: E\rightarrow S$ be a family of elliptic curves, i.e. a proper flat morphism of schemes whose geometric fibers are genus $1$ curves, along with a section $e:S\rightarrow E$. Then there is an invertible $\mathcal O_S$-module $\omega = f_*\Omega^1_{E/S}$ on $S$, sometimes called $\mathrm{coLie}(E)$ because at $s\in S$ the fiber of $\omega^{-1}$ at $s$ is the Lie algebra of the elliptic curve $E_s$.
I've seen it claimed that $\omega\cong e^*\Omega^1_{E/S}$, though it's not clear to me why this is true, or how even to define a map $f_*\Omega^1_{E/S}\rightarrow e^*\Omega^1_{E/S}$. I'd appreciate an explanation of why this isomorphism holds (or a reference giving such an explanation). Thanks!