Paul along with Timothy, wrote to Philemon, a wealthy Gentile slave owner from Colossae. One of Philemon's slaves, Onesimus, had deprived his master and run away to Rome/Ephesus to escape detection. Then he came in touch with imprisoned Paul and became a Christian.
Paul loved Onesimus very much - just like his son, but he was Philemon's legal property. Therefore, Onesimus should return to Philemon. Paul is writing this covering letter to Philemon to receive Onesimus as he would receive Paul. Tychicus is going along with him as a moral support.
Beginning with a thanksgiving, Paul opens his letter to Philemon. This is the only time Paul identifies himself as a "prisoner of Christ Jesus." He then prays for Philemon so that he may grow in his fellowship so as to understand what belongs to believers in Christ. Paul recalls how Philemon has refreshed the hearts of Lord's people.
Even thought Paul would have loved to keep Onesimus with him for his gospel related work, it is not right to do so without Philemon's consent. Perhaps there was a purpose why Onesimus escaped.
Philemon 1:15-16 Perhaps the reason he was separated from you for a little while was that you might have him back forever no longer as a slave, but better than a slave, as a dear brother. He is very dear to me but even dearer to you, both as a fellow man and as a brother in the Lord.Paul appeals to Philemon to receive Onesimus as he would Paul himself; if Onesimus has wronged, that is to be put to Paul’s own account. Paul asks Philemon to prepare a guest room for him, because he hopes to be restored soon.
Paul sends greetings to Philemon from common friends in Rome. The grace of the Lord Jesus Christ be with your spirit.