I got this set by accident on Dec 16th.
I was trying for nearby Alpha Pict!
I can't find any online source which mentions what the
faint nebulosity is immediately adjacent Beta Dor.
It's quite an interesting star in itself, one source says this about it:
http://stars.astro.illinois.edu/sow/betador.html
" Among the most important of all stars are the Cepheid variables, named after the prototype, Delta Cephei.
At its peak, Beta Dor actually reaches into third magnitude, 3.41, and then falls to 4.08, oscillating between the two over a slightly variable 9.842... day period."
Aladin DSS image also for comparison.
Steve