Paul says, because you have received grace, you owe me your own self. You have received grace through my ministry. You have some obligation to be gracious in return. He is saying here - I am confident that you will not refuse the claims of Christian brotherhood. He is still going to be your slave, you are still going to be his master, but he is going to now be so much more. And so you are going to now treat him differently, interact with him differently. You will do even more than that: not only receiving him back as a brother in Christ, not being begrudging, but going beyond what I require of you.
To quote from Muller,
Terry L. JohnsonPaul lays down principles, which in their logical consequence, finally would make it impossible to tolerate slavery in the bosom of Christianity.Not in a revolutionary way. Not the way the world wants, with headlines and wars and revolutions, but the indirect way—by transforming the relationship between master and slave. And by appealing to him as he would appeal to us in our relationships, that our relationships will be the relationships of love. They are brothers, they are family members, and so treat each other accordingly, as brothers and sisters in Christ. Treat each other according to the requirements of love. Fulfill the obligations of love for one another.1
20 Oui, frère, que j'obtienne de toi cet avantage, dans le Seigneur; tranquillise mon cœur en Christ.