Archie just went through this a month or so ago with a distemper shot.
It's not common, and Archie's never had a reaction like that before to
any shot, but lethargy can be a mild reaction to any vaccines and can
last, I think, up to a week or more after the shot.

General droopiness doesn't need treatment, or so I'm told.  However,
Archie stopped eating for several days and had some GI problems, too,
and I brought him to the vet.  His reaction was more severe, though it
wasn't a full-blown allergic reaction (which usually happens right
away).  You might want to just talk to the vet's office, so they are
aware of it and perhaps can tell you what to look for in terms of
symptoms of a more severe reaction that might appear and need to be
treated.

Barb

Yes, fairly common after a lot of different vaccinations. If the cat
does not show any signs of recovery within 48 hours then have a word
with the vet, obviously if he shows any real distress then it is
straight back to the vets.