doff (dof || dawf)
tr.v doffed, doffing, doffs. 1. To remove or take off:
doff one's clothes.
2. To lift or remove (one's hat) in salutation.
3. To throw out or away; discard. [Middle English
doffen, from
don off : don, to DO + OFF.]
We'll go for option
2, basically "doffing my cap" is like my acknowledgment of your abilities with Photoshop/Paint Shop Pro in virtually smoothing your bumper... another example of my slightly alarming eccentricities
It's not a phrase that is in common use any more, sadly...