In A harmonic minor the III is already a C, you can make it augmented but that is not very common. You generally change the v to a V and keep the VII diminished in both cases where as it is major in the natural minor.
He was saying that the III of C major is the V of A minor, and so if you are planning to play in A minor using a set of chords pulled from C major, you may want to add a III alongside the iii.