Haydn's long career includes familiarity with nearly all musical genres of his time. This broad background made Haydn a natural teacher and perhaps helped him to earn his nickname of "Papa." The selection we've just heard earned its own particular nickname from a lost divertimento titled The School Master in Love. One Haydn biographer also attributes the nickname to the dotted rhythms of the second movement, comparing it to the wagging finger of a disapproving teacher.
Haydn's most well known pupil was Ludwig van Beethoven. Haydn's influence is clear in Beethoven's early work, but even he couldn't keep Beethoven from forging his own path. Beethoven was often antagonistic toward his teacher, but Haydn still admired his student's compositions. The two were on good terms by the time Haydn's death.