After Paul's greeting to the church in Corinth, he writes about suffering and comfort through Jesus Christ. Timothy is with him as well.
The word "you" throughout this letter refers to the people of the church in Corinth and to the rest of the Christians in that area. Possibly Timothy writes on parchment paper the words that Paul says.
Your language may have a particular way of introducing the author of a letter and its intended audience. Alternate translation: "I, Paul ... wrote this letter to you, the church of God that is in Corinth"