Fairies become humans. The original humans go extinct.
Humans and Fairies are not equal in a fight.
Are each militarily roughly equivalent to an equivalently equipped and trained normal sized non-magical human due to their small size, quickness, proportionally greater strength and magical abilities.
Any fairy (are each) is as strong as a human soldier with full equipment. Guess what happens if you have trained fairies. They tear humans apart.
Fairies have a greater economical efficiency.
They require less space, can do more work and eat more. They're also more agile.
When faced with a power that is greater in both economics and military strength, you lose unless you can find a strength of your own.
But humans can't shun fairies from their society. Fairies are so small, they can live in the places where you didn't think to look. So fairies can't be wiped out, for they'll live everywhere. Humans don't have the means to detect them either.
It's not possible to end up with peaceful coexistence. Not even if both sides agree to such a thing. The population limiting factor for humans is economical and medical.
The economical constraint falls away for the fairies.
Short of a plague, there's nothing to inhibit fairy population growth. You've made them too strong. They have magic to help them in their tasks such as moving things, cooking things...
The fairy population will expand across the globe. Human territory cannot be defended and falls. Without education, you don't end up with very smart humans. You get inefficient working animals. Humans go extinct.
The only thing that would stop this is if fairies agree to leave certain areas alone.
Humans end up in a wildlife preserve.
Your question boils down to
"What happens if we add a"
- Stronger, and
- Magical, and
- Equally intelligent, and
- Environmentally resistant, and
- Flying
race of beings?
Well guess what. The previously existing race disappears.