I'm not a lawyer, and there may be some jargon here. But I would expect 'the subject of a contract' to to be actual words written at the top of a contract, and 'the object of a contract' to be the intention behind it.
Beware: there is a phrase - 'subject to contract' - where 'subject' is neither noun (stressed on the first syllable) nor verb (stressed on the second). When a house is sold (under British law), a 'sale is agreed' (the buyers say 'Yes, we want it', and the seller says 'OK,') and some time later (possibly months) 'contracts are exchanged'; that's the moment when the buyers own the house. But the agent ('Estate Agent' in Br Eng, realtor in Am Eng) doesn't want to wait a month or two before crowing over their success. So these notices appear outside the house, in this order:
- For Sale
- Sale agreed
The second can have - in tiny writing - subject to contract, either added on or in a diagonal 'flash'. (It's a fancy way of saying 'We're nearly there, just dotting the Is and crossing the Ts'.)
- For Teachers