Schaff, Philip. "Ch. 21: Apostles and Prophets." The Oldest Church Manual Called the Teaching of the Twelve Apostles. Second Edition, Revised and Enlarged. New York: Funk & Wagnalls, 1886, 67-73.
Schaff turns in his 21st chapter to consider the Didache's description of apostles and prophets. He considers that "the apostles spoken of in the eleventh chapter, are not the Twelve mentioned in the title, but their associates and successors in the work of Christianizing the world" (Schaff 1886, 67). He sees this as a secondary sense of apostles, missionaries who extend the Apostolate, such as Barnabas, Silas, Timothy, and many others (Acts 14:4, 14; 1 Thess. 2:6;' Rom. 16:7; 1 Cor. 15:5, 7; 2 Cor. 11:5, 12:11). They were seen as itinerant evangelists, hence the Didache's prohibition on receiving them for an extended period of time (Schaff 1886, 69).
Prophets, in distinction to Apostles, "were not sent as missionaries to the heathen, but instructors and comforters of converts, and might settle in a particular congregation" (Schaff 1886, 70). These were considered aking to chief priests, and were to be accepted and cared for in longer term settings. This ministry was gradually superceded by the episcopacy (Schaff 1886, 72).
Teachers also appear in the Didache, particularly in 13:1-2, where they have roles similar to both Apostles and Prophets (Schaff 1886, 72).