FWIW, I ran some F&F 3 Factor regressions for the S fund, Vanguard Extended Market Index fund [VEXMX], Vanguard Small Cap Index Fund [NAESX], Vanguard Mid Cap Index Fund [VIMSX], Russell Mid cap Index, and R 2000 index.
VEXMX is an excellent proxy for the S fund if places like M* and Vanguard can't x-ray it.
VEXMX and the S fund had higher small cap loading factors [SMB] than the Mid cap index and fund, but lower factors than the small cap index and fund. So, VEXMX and the S fund are just mid caps + small caps [roughly half of each].
An easier way to see this is to go to M*'s Portfolio snapshot of each fund:
You'll see that VEXMX has a lower Average market cap than VIMSX [mid cap], but higher than NAESX [small cap]. Hence, both small caps and mid caps.