Translation:He should not write on library books.
And how would 'lui non deve scrivere. etc' translate? I would say (direi) exactly the same. But it seems from a quick Google that the conditional is what people use for 'should not'. So why is that? In English, I suppose 'should not' is somewhat weaker than 'must not', so perhaps the italian conditional of dovere gives a similar nuance?
But dnovinc suggested above that 'lui non deve scrivere...etc' means 'he doesn't have to... etc', which is something quite different - it means, 'yes he can, but he doesn't have to', eg 'I am on holiday, so I don't have to go to school tomorrow' (I am released from my obligation to go to school). I mean here, he MUST not, he is obliged NOT to 'I have a highly infectious disease, so I must not go to school tomorrow' (I have an obligation NOT to go to school). Is there no way to convey these two different concepts differently in Italian?
Lui non deve scrivere = He doesn't have to write
more about conditional (with examples):