to Timothy, my beloved child: Grace, mercy, and peace, from God the Father and Christ Jesus our Lord.
Paul asserts his apostolic authority not for Timothy's sake alone; Timothy already believed. He writes with these grand titles because he stands at death's door and means to seal his whole ministry with his blood, that the Church may know his doctrine comes from God Himself, not from a mortal man's opinion.
AI summary
Commenting on 2 Timothy 1:1-18
my dearly beloved son--In Ti1 1:2, and Tit 1:4, written at an earlier period than this Epistle, the expression used is in the Greek, "my genuine son." ALFORD sees in the change of expression an intimation of an altered tone as to Timothy, more of mere love, and less of confidence, as though Paul saw m him a want of firmness, whence arose the need...
Verse 2. To Timothy, my dearly beloved son. . Grace, mercy, and peace, etc. . (b) "beloved son"