Do Monster Enhancers go with a Monster when it is removed from combat?

[Ukeeku]

My question, and I have searched for the answer, is;
When a monster is removed from battle, do the monster enhancers go with it?
Are monster enhancers attached to a specific monster in a battle? Or is it +10 to the monster side?

[MunchkinMan]

Monster Enhancers are played on a specific Monster. One-shot Items that are used in combat are typically used to give bonuses to a side, though some cards, obviously, have different effects.

To confirm this, the rules say this about Monster Enhancers (with my added emphasis):

Certain cards, called monster enhancers, raise or lower the combat strength of individual monsters.

These are the cards that say +X to Monster, which means an individual Monster, not the Monster side.

Furthermore, if you remove a Monster with Monster Enhancers attached to it, and that removal method gives Treasure, you get the Treasure that the Monster Enhancers add to the Monster.