Spectral sequence operations converge to Araki–Kudo operations

Previously, we constructed operations in the mod 2 homology spectral sequence associated to a cosimplicial E∞-space X. The correct target for this spectral sequence is the homology of Tot X. Noting that in this setting Tot X is an E∞-space, we show that our operations agree with the usual Araki–Kudo operations in the target. We also prove that the multiplication in the spectral sequence agrees with the multiplication in H∗(Tot X).