# You yourselves are our letter of recommendation

That the Corinthians have become believers serves to validate Paul's ministry to others. Alternate translation: "You yourselves are like our letter of recommendation" 

# written on our hearts

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. 

# written on our hearts

"which Christ has written on our hearts" 

# known and read by all people

"that all people can know and read"