In https://arxiv.org/pdf/1906.08616.pdf eq. 3.51 the following identity is proposed to hold for two-form $S$ on a manifold with metric $g_{\mu \nu}$.
$d \star S = s \cdot \epsilon$,
where $d$ is the exterior derivative, $\star$ the Hodge star operator and $s^\mu = g^{\mu \nu} \nabla^\sigma S_{\nu \sigma} $ and $\epsilon$ is the volume form of the given manifold. Is there an easy proof for this identity? Trying to work it out in coordinates became a mess to me and I couldn't find an reference for this identity.