You almost got it. But you can add name field constructor to your Object1 and try this code
map.entrySet().stream()
.collect(Collectors.toMap(Map.Entry::getKey,
entry -> entry.getValue().stream()
.map(Object1::new)
.collect(Collectors.toSet())));
与恶龙缠斗过久,自身亦成为恶龙;凝视深渊过久,深渊将回以凝视…