  In the present paper, we continue the research in \cite{Zh21c} to develop the
Sahlqvist-type completeness theory for hybrid logic with satisfaction operators
and downarrow binders $\mathcal{L}(@, \downarrow)$. We define the class of
skeletal Sahlqvist formulas for $\mathcal{L}(@, \downarrow)$ following the
ideas in \cite{ConRob}, but we follow a different proof strategy which is
purely proof-theoretic, namely showing that for every skeletal Sahlqvist
formula $\phi$ and its hybrid pure correspondence $\pi$,
$\mathbf{K}_{\mathcal{H}(@, \downarrow)}+\phi$ proves $\pi$, therefore
$\mathbf{K}_{\mathcal{H}(@, \downarrow)}+\phi$ is complete with respect to the
class of frames defined by $\pi$, using a restricted version of the algorithm
$\mathsf{ALBA}^{\downarrow}$ defined in \cite{Zh21c}.
