Why did the immorality in the Corinthian church bring reproach on the name of Christ in the city?
Because the sin was open and known throughout Corinth, even by pagans, yet the congregation remained puffed up and did not mourn or remove the offender. Their tolerance of gross immorality, rather than grief and discipline, exposed Christ's name to public reproach in the city.
Les explains that the situation involved a stepson and stepmother living together as husband and wife, a gross immorality even for Corinth. Paul condemned the Corinthians because they were puffed up instead of mourning, so that the offender might be removed from fellowship—not put to death, but taken away from among them. All the factions, whether followers of Peter, Apollos, or Paul, knew about it and were not bothered. Because even the pagans knew what was going on and it was not hidden, it brought reproach upon the name of Christ throughout the city of Corinth.