Paul begins an extended metaphor that speaks of the way that the Corinthians have been obeying Christ as if it were a letter that Christ had written first to Paul and his companions and then to all the people in the world. (See: [[rc://en/ta/man/jit/figs-metaphor]])
Paul speaks of the Corinthians as if they are a letter of recommendation. That they have become believers serves to validate Paul's ministry to others. Alternate translation: "You yourselves are like our letter of recommendation" (See: [[rc://en/ta/man/jit/figs-metaphor]])
Here the word "hearts" refers to their thoughts and emotions. Possible meanings are 1) Paul and his coworkers are sure about the Corinthians being their letter of recommendation or 2) Paul and his coworkers care very deeply for the Corinthians. (See: [[rc://en/ta/man/jit/figs-metonymy]])
This can be stated in active form with "Christ" as the implied subject. Alternate translation: "which Christ has written on our hearts" (See: [[rc://en/ta/man/jit/figs-activepassive]])