A.T. Robertson Commentary 2 Timothy 1:2

"to Timothy, my beloved child: Grace, mercy, peace, from God the Father and Christ Jesus our Lord." — 2 Timothy 1:2 (ASV)

Beloved (αγαπητο). Instead of γνησιω (genuine) in 1 Timothy 1:2. He had already called Timothy αγαπητον (verbal adjective of αγαπαω) in 1 Corinthians 4:17, an incidental and strong proof that it is Paul who is writing here. This argument applies to each of the Pastorals for Paul is known by other sources (Acts and previous Pauline Epistles) to sustain precisely the affectionate relation toward Timothy and Titus shown in the Pastorals.