Beloved (αγαπητο). With this vocative verbal (four times in this chapter), Peter "turns away from the Libertines and their victims" (Mayor).

This is now the second epistle that I write unto you (ταυτην ηδη δευτεραν υμιν γραφω επιστολην). Literally, "This already a second epistle I am writing to you." For ηδη see John 21:24. It is the predicate use of δευτεραν επιστολην in apposition with ταυτην, not "this second epistle." Reference apparently to I Peter.

And in both of them (εν αις). "In which epistles."

I stir up (διεγειρω). Present active indicative, perhaps conative, "I try to stir up." See 2. Peter 1:13.

Mind (διανοιαν). Understanding (Plato) as in 1. Peter 1:13.

Sincere (ειλικρινη). Old adjective of doubtful etymology (supposed to be ειλη, sunlight, and κρινω, to judge by it). Plato used it of ethical purity (ψυχη ειλικρινης) as here and Philippians 1:10, the only N.T. examples.

By putting you in remembrance (εν υπομνησε). As in 2. Peter 1:13.