I'm not sure what your > symbol indicates here:

"Chord: {function} > {quality}?"

Chord quality indicates the elements of a chord in isolation: major, minor, diminished, augmented, dominant, added, suspended, etc.

C-E-G is a major chord.  C-E-flat-G is a minor chord.

Chord function indicates the relationship a chord has to its key signature & musical context: tonic, subdominant, dominant, etc.

In a major scale:  tonic (I), supertonic (ii), mediant (iii), subdominant (IV), dominant (V), submediant (vi), leading / subtonic (vii) 

Example:

C (I) / Am (vi) / F (IV) / G (V)

The functions of each chord are:

C - tonic

Am - submediant

F - subdominant

G - dominant

Hope that helps!