Wipe the saddle with a damp towel and then a dry towel. You’ll want to remove any residual soap from the leather as soap left behind may damage the leather over time. Some saddlers recommend using only a damp cloth and no soap.
Residual soap left in crevices and folds will also hold grit that can eat away at the leather. Use dampened cotton swabs or the corner of a towel to get all soap out of tiny stitching channels and crevices.
Wipe the saddle dry. The next step is conditioning, and most conditioners work best on leathers that are a tad damp.