2 Corinthians 5:18-20 (ESV) All this is from God, who through Christ reconciled us to himself and gave us the ministry of reconciliation; 19 that is, in Christ God was reconciling the world to himself, not counting their trespasses against them, and entrusting to us the message of reconciliation. 20 Therefore, we are ambassadors for Christ, God making his appeal through us. We implore you on behalf of Christ, be reconciled to God.
"Reconciled Us," "Reconciling the World," and "Be Reconciled are from the Greek word katallassō and means "to change mutually."
"Ministry of Reconciliation," and "Message of Reconciliation," are from the Greek word katallagē and means "to exchange." It carries the idea of divine favor resulting from atonement.
Please allow me to place these definitions into the verses and paraphrase them.
God, through Christ, mutually changed us to Himself and gave us the ministry of exchange from divine favor as a result of our atonement.
God, in Christ, was mutually changing the world to Himself, not counting their sins against them
God has entrusted to us the message of exchange which is from divine favor as a result of our atonement.
Therefore, as ambassadors for Christ, God appealed to the world through us and we appeal to you for Christ's sake, be multually changed to God.