Jamieson, Fausset and Brown on 1 Timothy 4:13

13. Till I come--when Timothy's commission would be superseded for the time by the presence of the apostle himself (1Ti 1:3; 3:14).

reading--especially in the public congregation. The practice of reading Scripture was transferred from the Jewish synagogue to the Christian Church (Lu 4:16-20; Ac 13:15; 15:21; 2Co 3:14). The New Testament Gospel and Epistles being recognized as inspired by those who had the gift of discerning spirits, were from the first, according as they were written, read along with the Old Testament in the Church (1Th 5:21, 27; Col 4:16), [Justin Martyr, Apology, 1.67]. I think that while public reading is the prominent thought, the Spirit intended also to teach that Scripture reading in private should be "the fountain of all wisdom from which pastors ought to draw whatever they bring before their flock" [Alford].

exhortation--addressed to the feelings and will with a view to the regulation of the conduct.

doctrine--Greek (ministerial), "teaching" or instruction. Addressed to the understanding, so as to impart knowledge (1Ti 6:2; Ro 12:7, 8). Whether in public or private, exhortation and instruction should be based on Scripture reading.