The book of Philemon is written by Paul (v. 1) between 60 and 62 AD during his time in prison (v. 1) in Rome. The letter is written to Philemon, Aphhia, Archippus and the church in Philemon’s house (v. 2). Philemon was a part of the Colossian church (col 4:7-9).
The reason for Paul to write this letter is to appeal for Onesimus. Onesimus was Philemon’s slave but was departed from him for a while (v. 15). He had probably run away from Philemon, had been useless to him (v. 11) and may have owned him something (v. 18).
During Paul’s time in prison, Onesimus had become precious and useful to Paul (v. 10-11). He became a believer (v. 10) and served Paul in his ministry (v. 13). Now, Paul is sending Onesimus back to Philemon. He asks Philemon to reconcile with Onesimus and to receive him as he would receive Paul (v. 17). No longer as a slave but as a beloved brother (v. 16-17). Though Philemon has the right to treat Onesimus as a slave and even punish him for running away, as the penalty for rebellion was crucifixion (Kerr, 1022), Paul asks him to treat him as an equal.
What stands out in this letter is that Paul leaves the choice up to Philemon. He doesn’t force him to forgive and accept Onesimus as his brother. He prefers to appeal to him (v. 9) but also speaks out the trust that he is confident of Philemon’s obedience (v. 21). He has heard about Philemon’s love and faith towards Jesus (v. 5) and how he has refreshed the hearts of the saints (v. 7). This shows something of the character of Philemon and seems to give Paul the trust that Philemon will make the right choice in this situation.
Kerr, W.N. ‘Slavery’. Evangelical Dictionary of Theology. 2nd Edition. Editor Walter A. Elwell. Grand Rapids, Michigan: Baker Academic, 2001, pg 1112-1113.