  The logic $\mathcal L^1_\kappa$ was introduced by Shelah in [3]. In [4], he
proved that for a strongly compact cardinal $\kappa$, it admits the following
algebraic characterization: two structures are $\mathcal L^1_\kappa$-equivalent
if and only if they have isomorphic iterated ultrapowers via $\kappa$-complete
ultrafilters. We give a presentation of the logic $\mathcal L^1_\kappa$ and a
simplified and slightly modified proof of this result.
